I like quiet horses with a bit of character to them who are well broke to saddle but haven't learned anything besides the basic walk, trot, canter, turn. I don't like silly horses, or those with not much personality.
My pet hate in horses (mainly lesson horses) is the ones that are really dead to the leg. They are just so hard to do anything with. You can't even focus on your own position because you're fighting to get them to move. And any improvement you make in your hour lesson is gone by the next time you ride.