Best and fastest split pushers?

Tryn is a good split pusher because you just can't ignore him. You also usually can't just send 1 person to go deal with him.

Nasus isn't a fast split pusher but once he gets his stacks up he demolishes towers and inhibs.

Shen doesn't really have the abilities for it, but if you build Lightning Shen he can do pretty well and he has constant map resence via his ult.

Mid wise, Heimer & Azir are your major pushers. Heimer can just leap frog his turrets up (although this is pretty un safe for him to do) whereas Azir can burst down the tower and then put his passive on it, shortening the minion's lane and keeping it pushed forward.

Sivir used to be the ADC queen of split pushing; her boomerang and ricochet can easily melt waves, and her ult can even get a creep wave sprinting to the tower.

Trist is the best ADC tower killer right now, hands down. Throw E on the turret and press Q for atk spd (which make the E bomb do more dmg) and that turret is gone very quickly.

Firstly, it is uncommon for ad carries to split push since they are usually need to stay with other members. Top lane, jungle, and some mid lane champs will be the ones to split most commonly.

The fastest one in each role imo:

Top: Tryndamere Mid: Zed Jg: Yi