tl;rd first - Katarina is OP.
Basis of this rationalization is lolbase data, only v188.8.131.52 games, only US server, only lvl30 players, only 5v5 games. Amount of Katarina games on the record which meet that precondition is 20657 atm, which is more than enough to clear all corner-cases and assume that average case is measured correctly.
Chapter 1: Why Katarina is OP, statistical rationalization.
First, finally, Katarina is not most played champion anymore. After the patch attention switched to buffed Kayle, Olaf, Ez and Shaco, which, along with ever-popular Ashe makes Katarina #6 most used. Still, considering the company she got, no illusion about her tier should remain.
Next, she went down to #4 killer instead of #3, falling behind Karthus by 0.2 average kills per match. Still, #4 killer is #4, a single digit ranking.
Next, #3 in k/d ratio proves exceptional survivability for an assassin. The only 2 above are Ez and Shaco, with obvious reasons why their survivability as assassins is great. The 2 guys behind are Kassadin and Karthus, one because he's basically the same type as first 2, and second because he doesn't need a contact to kill people at all. Anyway, #3 hit'n'runner is Katarina. Another single-digit ranking.
Another single-digit ranking is her farming ability. Some have speculated that it became harder to farm with shorter range bb. Indeed, Katarina had fallen from #2 farmer to #5 farmer... with no falldown on average gold per match! It's just heimers, tryndameres and karthuses suddenly learned to farm better, so now there's 5 champiojns with 10k+ gold averages, and these is only 1 assassin among them. Guess which? #5 farmer spot, same amount of gold per match as in pre-nerf bb range case, another single-digit ranking.
Wait, there's more. Although amount of utility killers with positive both w/l and k/d ratios went from 8 to 12, katarina raises from #8 to #7 as a win-bringing killer, as her w/l ratio went 6% up from the last patch. That only means we get less dual-katarina matches though, so not every katarina win is accompanied by katarina loss, boosting her w/l ratio in a passive way. Still, #7 utility killer.
So, usage, killer, farmer, utility, survival - 5 single-digit rankings in one champion. On a statistical grade, this is OP.
Admittedly, these numbers show Ez is more of a problem than Kat is, but unlike kat he isn't a farmer and has another price to pay as assassin for his power. More of it in Q&A chapter. Also, Ez is a known and confirmed problem, making his discussion unfitting for a problem definition thread.
Chapter 2: Why Katarina is OP, mathcrafty/theorycrafty rationalization.
Well, that one is simple. Katarina has #1 raw damage ability in the game. When calculating power of her ulti, most people neglect the point that daggers are affected by KI passive, granting them 24 extra damage when KI is maxed out, making it able to do 660 (lvl3 KI) raw damage at level 6 with no items/runes/masteries and only 1 ability involved. Now if you add those...
Disregard that, problem is not her ulti damage. Problem is, that damage from her other abilities can get you low enough to kill with ulti even if it did less than ignite in terms of damage. Her abilities are unkitable, be it bb or shunpo, since staying out of range is only an option for ranged champions, and you cant outrange both shunpo and BB with almost anything, maybe except Kog's artillery. That means you will get hit. Healers will get their healing reduced, regenerative champs will get their regeneration cut in half and handed to them, so they will regenerate less than Annie, while wearing regrowth pendant.
Laning against any decent Kat is facing a 1000+damage challenge every time you go for a lasthit, as there is exhaust/ignite, #1 damage ability, healing reduction and free positioning skill with some damage waiting for you. That isn't kind of laning power assassin should have. Throwing in a farming power this particular assassin has, and you get OP picture. Throw in a survival power of a tank-class base HP (close to Singed, more than Malphite, Shen and Cho's base HP at lvl18) and damage mitigation ability, along with low-cd free escape capabilities, and you get very OP picture.
Chapter 3: Why Katarina is OP, Q&A rationalization.
Q: Why don't you stop her ulti with CC?
A: As discussed ton of times, it's barely possible in most situations. Let's do a listing (yes, again) of champions who can stop Kat's ult, but give it a few restrictions. First, it's got to be a ranged ability. Walking in with 400 movespeed on 600 units range ability will require 1.5 seconds, which means taking most of the lotus and perhaps some punishment from katarina's teammates, which imo disqualifies non-ranged abilities. Second, that must be an ability which can be landed regardless of obstacles. Minions or enemy champions in the way will break your kat-stopper in a teamfight, unless they or kat are bad, by just waling in between you and kat, which also disqualifies skillshots. Third, no preconditional abilities, because they're not available every time you want them to, disqualifying Annie and Kennen. Well, it's not like those two had a fitting role anyway, since they're nukers and supposed to nuke enemy dd, not stand still waiting for kat to jump and doing no damage. Moreover, I'd like to disqualify abilities which have more cd than kat's ult, because they're guaranteed to not be available every time she spins. Now let's see what's left:
17/54, roughly 1/3 of champions can do it. Now, these 17 are seriously gimped by Katarina. Why? Because she took away their CC! Now they can't initiate with it, can't waste it to get a kill, that's literally like they don't have it at all, until kat spins!
Now let's assume kat got a bveil and not stupid enough to let it pop before starting to spin. How many champons have 2 CCs of the same type to be able to stop kat after having first one busted? Busting it with other spell can prove difficult, because you see, when katarina starts to spin it's already too late to carefully think it over, and ability is used and pops bveil. The answer is... 1 champion. Fiddlesticks. And maybe Malzahar, if you go against restrictions to add his long-cd ult to it.
So, basically, "stop her ulti with CC" means either "Play on a newb island" or "Play fiddlesticks all the time and don't use your fear and dark wind EVER before you see kat spinning". Both of those are barely acceptable suggestions for everyday gaming, which concludes the question.
Q: Why don't you just walk away from her ulti?
A: That is an option if she missed a spin and you have a short way out of DL circle. If that isn't so and spin is placed correctly, you end up taking most of its damage, even with 400 movespeed and uncced. However, if there's a Rylai, exhaust, or (in most cases) Kat has a cc-capable partner with her(which can be anyone with exhaust), you're getting it all and extra kick in the balls from partner.
Another situation is a teamfight, where even if you do walk away, some of your teammates wouldn't. That brings us to "run with a whole team from 1 champion" as the only possible solution on a walk-away countering basis, which is plain sad if you ask me.
Q: Some champions have a good escape ability, tried that?
A: Aside from the fact that not all of them do, what do you accomplish by running away? Considering how low CD on DL is, you have to run away from all teamfights for the whole game. And that's the only ability of kat you can run away from. Now remember there's 4 others there...
Q: So you can't stop Katarina at all?
A: Of course you can. That only takes a 5 man premade of experienced and heavily communication-aware players with a specific katarina-countering lineup... to shut down 1 champion. Since there's 4 others out there, lineup is narrowing up, coming out with a little to no variability, boring your team to death with itself. While it is admittably efficient way to counter Katarina, it is not an answer to winning in general, and even less of an answer to having fun while you do.
Q: Katarinas don't get bveil, do they?
A: Well, that has a certain point. Let's take a step aside and look at Nunu. Fairly similar ulti, and bveil as an obligatory item. Why Nunu take that one as obligatory? Is it for hitpoints? No, Nunu base HP is among the top. Is it for mana? No, tear of goddess will give you many times more mana at many times less cost, and add mana regen on top of it. Magic resistance? Abyssal Scepter offers same MR along with AP and MR reduction, Force of nature offers more MR, and both of those are cheaper! Let's be straight and admit Nunu have to take this one because of its passive. Now, back to Katarina. Her base HP is tank class, but not among the top, so if anything, she will make use of extra HP. Mana is a waste, but bveil is clearly not a mana item(as there are much better choises mana-wise), and its passive is just made to protect this "oh so stoppable" ult from most stoppers out there. Means if Katarinas don't get it, they deserve what's coming. Otherwise, if they really take it as rare as some people see it to be, that might mean being stopped mid-spin is not a problem for them. Because if it is, you'll get a bveil, right?
Q: Katarina is squishy, why don't you just focus her?
A: Confirmed to be non-true. 1972 base HP at level 18 is a tank class base HP. 30% damage mitigation ability for 3 seconds is a tank class ability as well. Look at Shen. 1958 base HP at level 18 and 240 HP feint equals 12% mitigation ability(at best), which only lasts 1.5 seconds. Is Shen squishy? No. Means Katarina isn't either. Now about focusing her... what's the point? Before she spins there is a better targets, like Ashe or Veigar, strong damage dealers with CC abilities. After she spins - there is no difference if she lives or not, she's already done her worst. And most importantly, she still has her free escape, so all your focusing has a good chance of being screwed up. And if you will focus her, she goes down with 3-4 assists, combined with her passive it's a fortune on its own, which means it's better to focus on surviving than on killing someone who did her worst already. So, to sum it up, Katarina is not squishy and you have a reason to focus someone else any moment between before spin and after spin, and you can't do it mid-spin because of game's #1 damage he deals to you in that period.
Q: Magic resistance lowers magic damage from bb, shunpo and dl. Tried it?
A: You bet I did. Unfortunately, there is a thing called "magic penetration" in the game. Katarina doesn't need mana regen, cooldown reduction (passive takes care of that hard enough), which is important for all other casters (except maybe kennen), leaving magic pen pretty much the only thing she needs to unleash her #1 damage. By using pretty standard build with boots and guise she can already almost to 60+15% pen, buying void staff if necessary she will completely negate your (costly and not fitting any champion) MR-stacking attempts while not straying too much away from her build for more than 1 item, which makes this soft-counter barely efficient unless you're playing a tank and stack a ton of MR not straying away from your build as well. Have you seen 5 tanks lineup ever? If this question suggest we should try it to counter 1 champion, my answer should be that this champion is probably OP.
Q: Ok, she is strong, but not OP.
A: If your roster consists of "tier 1 only" and you're likely to post in "RIOT killed *" thread every time someone getting demoted from tier 1, you might feel that way. Or if you're a fiddlesticks player all the time, regardless of this champion's current tier standing. There is a dozen and half of overally viable champions in tiers 2-3, which get they butt handed to them by katarina player rolling face on keyboard. Unless you never play those, I cannot see how that opinion may form, except a not-as-rare-as-it-should-be case of you being an emotion-driven katarina player.
Q: Aren't you just qqing about being raped by Katarina players because you're just bad?
A: Did I say somewhere I get raped by Katarina? Not at all, if Katarina player doesn't know his champ as well as I do, he will become food. Then again, a half-decent Katarina player is a game-breaker for me, because it's totally not fun to play against. He's almost impossible to gank, you get outdps'd in any 1v1 situation, including laning, and if you don't work with a whole team Katarina will snowball out of killing champs 3 levels below for a full gold reward (hi passive) into a fed unstoppable qq-generator.
Katarina has no real counters among items and summoner spells, the only half-decent suggestios are MR items (discussed already, 55% + 60mr is neglected, making it barely half-decent) and Flash(having 4x Kat's worst cd time cooldown, quarted decent at best), only thing what's left is champion abilities. I remember once I made enemy singed QQ about my Yi wearing bveil, every time I got squared with singed with bveil up I just popped meditate and saw how fling breaks bveil and singed being unable to do anything but poison 400mr and regening Yi after that. That is the end of all hope if you only have 1 CC of required type on your team and kat gets a bveil, as well as end of all hope if you stack MR and kat gets a void staff. She can outdo you at any attempts to counter her, and even if you do counter her with aforementioned communicating experienced team and katarina-countering lineup, she still ends up gimping your team by taking away your CC with threatening to press R.
To sum up this answer, bad kat gets raped, while good kat is bringing too much utility on her team's side, making gameplay exhausting and variationless just because there is a katarina in play.
Q: Did you even tried to play her yourself?
A: 12 wins 1 loss, never less that 7 kills per game. Loss was a second game, where team fed fiddle. All solo queue.
Q: Speaking of Fiddlesticks, isn't he overally better than Katarina?
A: No, he isn't. Damage on his abilities isn't that great(avg damage per match, katarina 98160 Fiddle 81937. Considering both do damage with pretty much only abilities...), he's much less of a farmer(gold per match, kat 10069, fiddle 8467), less of a killer(avg kills per match, kat 7.51 fiddle 5.50), somewhat less of a surviver(avg deaths per match kat 5.27 fiddle 5.64), and the only kat-like point of him is his ulti damage. Being close or even more than a Kat's, let's not forget it has 180 seconds cooldown, gives a 2 second warning before activating, and, unlike katarina, fiddlesticks IS squishy, with 1830 base HP at 18 and no damage mitigation ability. He, of course, can try draining, but at a cost of mobility, which is vital because of a smaller radius than that of a DL. Overally, I see no statistical advantage on fiddlesticks over katarina. The only point of fiddle being able to counter katarina is not enough to declare him comparable.
Q: Why are you trusting lolbase so much, it's not a full picture.
A: 20000+ matches of only lvl30 players for katarina alone rule out possibility of picture being unclear. A level 30 player is a player who has at least 200-250 games, enough to get a clue on what's happening. 20000 katarina matches means there are 200k individual champion plays involved in this data, solid enough number for me.
Q: Lolbase players aren't all high elo.
A: You aren't either (me too), top 500 elo list population aside.
Q: So, why Katarina is OP, any tl;dr version?
A: Because she do too many things very good, have game-wide #1 burst damage, free escape ability, tank-class base HP and damage mitigation, Sivir-class minion farming ability with extra gold from passive and kills, and no widely available counters except specific champions and disadvantageous usage of abilities on those specific champions. All those for no cost and at a joke of cooldowns.
Well that's simply to huge mate. Sure some may read it in entirely but making your writing more concise would vastly improve the number of people who comment.
As for my stance on Kat. I think she's fine. She does a few things decently well and 1 thing exceptionally well unless CC'd. She's decent at farming and harassment/laning but can't really knock down a tower with that horribly slow attack speed of hers. Her ult lacks heavily in team fights mid game simply because he doesn't target many people. And while her ult is very very strong it's a CC magnet.
If anything I'd like to see Kat have her ult weakened and her other abilities made more powerful so that she's less of a 1 trick pony. As it stands currently by late game she's kind of like old version Kayle. A walking ult and some other annoyances.
In my book Kat's perfectly fine and based off of what Shurelia is saying (ie: what the offical stats, not lolbase) are saying about Kat is that she's indeed fine. I wouldn't expect changes if I were you.
I still remember how much people cried when Gangplank's grog soaked blade applied on his parrrley hits. Strangely enough bouncing blade does the same thing without a mana cost and with the added ability of being able to hit multiple targets. Sure, it might not do the same amount of raw damage per hit but unlike parrrley it's completely spammable. This thing will shut down a heal focused composition.
So yeah, I agree with most of the critique. A simple solution would be giving her a mana bar.
Stats aren't everything, some heroes don't need kills or farming to be game breaking, some heroes initiate then die, some heroes support etc.
While im not saying you are wrong, here are some things to consider
-Kat is a one trick pony, for all that farm shes getting, its all required for her R to be successful whether to enhance damage or increase survivability. If she didn't farm that much, she would fail. e.g. id rather have my kat farming than my fiddles.
-Also consider Kat's role in a team, in team fights there are acceptable champ deaths, tank, supporter. Kat isn't one of them, the other 4 heroes set up the ult for her, and she kills people with lower risk. If they they couldn;t set her up, well shes failed again and won't do much in a fight.
- Her auto attacks are poor, only thing she provides is her ulti, and BB. BB with reduced ranged has balanced that somewhat.
Basically, all her farming capacity, low deaths (play carefully, shunpo), her damage mitigation all attribute to her one trick, ulting. Her ult would fail if those things weren't in place, then she'd be useless.
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