Ahri

Ahri is a pretty easy lane pre-6, since she has to cast an animation before throwing out her charm, which makes it easy to dodge with W and/or Shunpo. After 6, Ahri's multiple dashes with R make her a bit more deadly, as she can potentially dodge most of your daggers and outplay you.

Akali

Level 1 Akali heavily out damages you.
Level 2, you out damage her if she takes shroud. If she takes shuriken she'll win any trades.
Level 3 and beyond is a skill matchup, typically if she shrouds the trade will be favored to her so bait it out before fighting her or immediately disengage if she uses it in the middle of a trade.
Post 6, you can ult in the middle of her shroud and it'll still hit her, giving you a large advantage, as if she stands in your ult she'll take very heavy damage if she doesn't have the items to burst you down herself.

Annie

If she has less than 2 stacks then you can trade. Post-6 she can throw Q at you and then drop Tibbers, so don't fight her when she has at least 2 stacks. If you're feeling uncomfortable about this matchup, picking up a Hexdrinker or taking Nullifying Orb from the Sorcery tree may help you out.

Aurelion Sol

Azir

I've yet to go against Azir enough to actually know how to play against him, but be aware a good Azir will make you want to die.

Brand

You might want to take Doran's Shield against him.
Early boots would help with dodging his AOE.
He can spread his burn from nearby minions to you if they are in range, so be wary of that. If you see him start throwing his AOE, Shunpo on him and W. He'll likely use his point and click on you, then throw his stun at you, so you'll have to anticipate the stun hitting.

Anivia

Diana

Don't run away from her Q, dodge to the side opposite of where it arcs. Diana hard wins this matchup at all stages of the game so be wary and only fight her if she wastes her shield or Q on the wave.

Ekko

If he misses Q, then you win all trades against him (pre-6). After 6, be wary of his ult and fighting him. Don't ever stand on where his ult indicator is standing.

Fizz

A good Fizz will never let you deal damage to him that isn't Shunpo or autos. Only fight him is he uses his hop and dash on wave, as without those he is his weakest. Buy an early Hexdrinker and always be wary of his ult.

Galio

The only way to win lane against Galio is to try and get a lead on him as early as possible. Kill him level 2 and lane should be set.

Heimerdinger

Pre-Gunblade you can't really do much other than ping your Jungler for help or roam to another lane to get a lead. Just be wary of his stun, and try to take out his turrets when they're targeting minions.
Don't be too nervous about being shoved under tower, there's not much else you can do about it, other than farming what you can.

After Gunblade you should be able to 100-0 him.

Lux

Lux is annoying but this lane should be easy.
As soon as she throws Q or E at you, you should be able to Shunpo onto her, effectively dodging the damage and winning any trade.
You can go on the offensive as well by Shunpo'ing onto her and W'ing around her to dodge the root.

Nautilus

His ult is a point and click knockup. Pick up a Banshee's Veil if you really need it, otherwise Zhonya every ult he throws at you.

Cassiopeia

Her ground makes you unable to Shunpo, so be careful when jumping onto her.
If you engage her after 6, make sure you turn Katarina's model away from Cassiopeia when you ult because she can easily ult you.

Zed

Picking up an early Seeker's Armguard and/or Stopwatch may be optimal against Zed. If he ults you, either use Stopwatch, or blink away as soon as he reappears so that a majority of his damage will be mitigated.

LeBlanc

You can dodge her dash by jumping away from where you're standing, Often Leblanc's aren't expecting you to dodge it, and it mitigates a lot of her damage.
Aside from that, she typically outdamages you and always out-trades you early. If she tethers you, disengage immediately by Shunpo'ing out of it's range, or W away from her. You can also get a reset by killing her passive clone that spawns.

Neeko

Dodge the root and all should be fine. If she starts ulting, you can Shunpo out of it before re-engaging her to fight.

Yasuo

Windwall blocks Q dagger from spawning so be careful of that. He can also completely block your ult.

Jump on him as he throws a tornado out and W and you should win the trade.

When he doesn't have any minions to dash to, he's also much easier to deal with.

Veigar

He's annoying, but completely harmless. Your entire kit counters his, so you should always win this. Shunpo allows Katarina to escape his cage and onto him.

Talon

He can 100-0 you level 2-3. Talon is notorious for having a very scary early game, so honestly rush a Seeker's Armguard if you need to. Respect his early game, it's a heavy skill-based matchup in Talon's favor.

Kassadin

You need to kill him as early as possible. Level 2 and 3 will be your main opportunities to, because he ramps up very quickly. Always take Conqueror and Long Sword into him to kill him as early as possible and run him down with autos.

Yuumi

Killing Yuumi's host almost always leads to a double kill. Her poke is easy to dodge and her healing is greatly reduced with Katarina's ult. Just beware of being hit with Yuumi's ult, otherwise she has no peel for her team.

Yorick

Taking him on 1v1 will almost always result in your death if he has a maiden and you're not fed off your ****. Only fight him if it's a 2v1 or more. Beware of his skillshot, which will make all of his minions jump towards you.

Xerath

Xerath has so much burst if you get hit by his stun you automatically lose the trade. Pick up tier 1 boots early if you're struggling to dodge his poke.

Vladimir

You need to kill him as soon as possible (Level 2). Once he gets his pool at level 3 the lane will be much more difficult.

Soraka

Her silence will make you want to die. Don't rely too much on Shunpo for movement against her as you'll be helpless when she silences you.

Sylas

Do not ult if he still has his knockup. Do not get combo'd level 1.
Otherwise you should be fine.