If it's an AD champion, Jax all the way. His ability to dodge, stun, and stick to a target, all while putting our massive amounts of damage and becoming tanky put him way ahead of anyone who relies on auto attacking. I love when I'm playing Jax and someone tries to duel me, they really don't have a chance. If it's AP, I'd say Kassadin. His passive soaks up magic damage, the burst he puts out is insane, he has high mobility, and he has the ability to silence his opponent. That's really hard to beat.
@Twang, Yorick wins any long drawn out fight as nobody can out harass or out sustain him. However, in a committed 1v1 he's only so-so. He pops his ult and unless you got in Q range you kite him until his ult ends, and if he's still committed to fighting, you can kill him at that point. If someone tries to engage on him though, yes he's very strong at 1v1ing.