Hello, I'm Katariven, a Gold Katarina main. I've stopped playing frequently since the start of the current season, but I've gotten back into it enough to notice some things about the state of Katarina and the game in general. I believe I have some good math (and builds) to contribute to the community. You can play Katarina in midlane, top, or support. She's generally picked midlane, because she's very hard to gank despite being squishy if she has her E up.

I always run my Gyrados set. Magic pen reds, armor yellows, MR blues, and MS quints.

Always run flash ignite on katarina. Maybe if you were toplane, you could do teleport and if you were support then you could do something different, but who knows.

Now on to explaining the inclusion of Hybrid kat.
Let's think about items first. Obviously you will not build hat. Mercurial Scimitar becomes a very nice option for a defensive slot, replacing the now-inferior Banshee's or GA. Katarina can still benefit equally well from items like Ludens, Abyssal, and Zhonyas. Of these, Ludens tends to work the best for Katarina's style. Lich Bane also gains value, even if the mana is a waste. There's not many strictly AD items that you would want, because you just want AD, AP, and magic pen, not many AA related effects or stats. Maw is pretty good if you're facing AP casters or assassins.

Let's consider two full builds and ignore the journey to them outside of raw cost. We can also mention that the hybrid build has an easier path because you get to cut out a needlessly large rod.

We're also eliminating void staff despite it being core to both builds, because its effectiveness is dependent on the enemy itemization.

AP Kat: Sorcs, Ludens, Hat, Abyssal, Zhonyas, Banshee's. COST: 15990
Hybrid Kat: Sorcs, Rageblade, Gunblade, Ludens, Scimmy, Zhonyas. COST: 17200
Hybrid costs 1210 more. A bit, but not too much. Now for the pro-cons.

Hybrid Kat is safer if you use the actives properly. You can actually jump in and do part of your combo, autoattacking some, and then scimmy off the stun that hits you before ulting. Both builds have the zhonyas option. AP kat has the banshees which is better against something like thresh or blitz.

AP kat has= 559 bonus AP. 50 armor, 110 MR. Banshee's and Abyssal passives. Bigger ludens proc.
Hybrid Kat: 392 bonus AP (With stacked rageblade). 150 bonus AD. 50 armor, 90 MR. Scimmy active. Rageblade passive. Gunblade active. Spell vamp and lifesteal from gunblade.

Kat has ratios of 125% AP 60% AD on her QWE and 250% AP 375% AD on her ult. But you won't get all of that off. Let's say you channel half the ult and that it hits two targets. Fairly average result, right?

Disregard runes and masteries now.
Ludens will proc for 184 on AP kat and 159 on hybrid.
QWE will deal 575(base)+699(AP) magic damage on AP kat and 575(base)+490(AP)+90(AD) on hybrid kat.
Half an ult will deal 375(base)+699(AP) on AP and 375(base)+492(AP)+281(AD) on hybrid kat.

So damage-wise, your basic combo will deal 1458 (with ludens) on AP and 1314 on hybrid. Slightly less damage, okay.
With half an ult though, hybrid will deal 2462 to AP's 2532. An even less noticeable difference. Gunblade active pushes both combos over the threshold, dealing 157 damage and making hybrid more damaging in either scenario.

Clearly this isn't enough of a reason to run hybrid though, so what ELSE do you get? Well, instead of blocking a thresh hook, having 20 extra MR and lowering enemy MR by 20, you get a LOT.

You can cancel a stun with scimitar. You get 20% spell vamp and 10% life steal. You get a SLOW. And somewhat irrelevantly, you get 32% attack speed. PLUS when you fall under half, you get 20% more attack speed, and 10% more of lifesteal and spellvamp. All this for an extra 1210 gold. So much more utility, and when I play katarina, the thing that I find to be missing is sustain. I'll have to back with 3/5 health because I simply have no way to gen it back and I don't want to be caught or have to fight like that.

No, support Katarina is not good. But I used to be in a dark place, where all I could play was mid and ADC. When I got stuck with support, sometimes I'd pull it out instead of being resigned to get **** on as thresh. It's cheesy, and not great but it's good enough to **** on people that don't know what they're doing below high gold. You can play worse champions in the support role. What I will offer, however, is that you do not play as a traditional support. You play hifey as ****. You will be the one getting kills. Play to become a threat early on, and get your carry ahead in farm. But take the kills. The whole point is that you're a cocky prick who doesn't need the ****ing meta to tell him what to play where, and that you're good enough to succeed anyway. If you do manage to get ahead, your enemies will be incredibly pissed. They will flame each other and you will win easily. People cannot STAND being beaten by weak or cheesy builds and champions, especially not at low ELO.

I've found that sometimes it's useful to rebind some keys for kat. I currently don't play with either of these mods, but at points in time I've changed my trinket key to 2 and put wards on 3 for easy ward hopping. Basically you strike the ward key with your middle finger and then the shunpo with your index.

Alternatively, you can try swapping the E and W, since Katarinas combo is QEWR. This allows you to roll over your keys more rhythmically. It was more useful in the DFG days, and I'd bind item slot 1 to Q, shifting over all of my keybinds by one. So when I wanted to go all in: QWERT, wait, and then WER to clean up additional targets.

If these sound too weird then you don't have to try them. I don't currently use either mod, but I found them both useful at a point in history, so I've included them for the more adventurous.