I haven't bought another horse purely just for myself in years, but if I were to make a list on what I'd look for...
One thing that comes to mind is a horse that isn't too timid/jumpy/shy/spooky.
It's one thing if the horse hasn't been exposed enough, green, or was abused at one point, but it's another thing entirely if that's part of the horse's personality. I've known horses that always spook at that same **** bush on the trail, always shies away from your hand, etc.
I'm 5'10", so no ponies for me.
I like the Anglo-Arab built with the TB as the dam. I have never met an Anglo-Arab I didn't like, as long as the mother was the TB.
That tall, up-hill built is what I like. There's obviously other horses built like that, but Anglos come to mind first.
Personality. It's up there with the most important things, and the one I look for first.
I love those in-your-pocket type horses. The ones that are always eager to learn, and have very expressive personalities.
Those curious little schmucks who always find a way to open their stall door or pasture gates, carry around their buckets or anything else they
Put in their mouth, the ones who chew on the cross-ties, lead rope, or anything within reach.
A horse that is willing to go for a run, or go for that second trail ride.
I don't show or really go anywhere besides my barn. I just trail ride really, but I'd like a horse with some "go". Not one of those "tries-to-bolt-every-five-minutes", but will speed up when I say and still listen to me.
I don't care if the horse is registered or has good bloodlines.
No bad habits/health problems.
I will never own a cribber. I will never own another horse with heaves.
I want an easy keeper.
I wouldn't rule out a horse who has a club foot though. I've known a few that were still very ride-able with corrective shoeing.