All the things I hate in a horse (many mentioned already here) seem fixable...my mare used to walk off at the block, now she doesn't. She used to jig all the way home on the trail, now she doesn't. She used to throw her head up to avoid the bit, now she doesn't. She used to never stand still, like if you wanted to stop and chat on the trail, now she stands. So my peeve would be a horse that doesn't catch on to training quickly - I like a smart horse that might have a bad habit or fear issue or nervousness or whatever, but once you've shown them how you want something done they go "oh, I get it" and make a good effort and improve quickly.

My old horse was one of those random spookers - hate that, too. You'd be walking along and out of nowhere she'd take a huge leaping spin. PIA!