I choose my horses mostly by watching competitions and trying to judge a horse's ability and if I think we would work well together. It also really helps to have people you can trust that know the kind of horse you want and could work for you. This is not an easy process, and no one gets it right every time! There are some key qualities I look for, such as ability, acceptance of training, and natural quality.
There are some things though that separate the best horses from the good ones, and this I struggle to see but I believe one can feel. For example, when I tried Super Sox, there was a connection and quality I felt in him that I could not see from the ground. This desire and spark I don't think you can train into a horse, but rather some just have it.