There are several requirements.
Correctness. Crooked legs, poor feet, bad angles, walk away. I can forgive some things but not basic correctness.
Hind leg action. I walk way from any horse that is short stride or lazy behind. That is where the jump comes from. Being a bit choppy at the trot can be fixed. I like a good shoulder but I will forgive this before I will forgive a weak hind end. I do not tolerate the latter.
Brain. A horse does not have to be quiet. But I do not tolerate one that is mean in any way, as in kicking out, swishing tail, etc. I recently walked away from a very handsome horse that tried to kick my vet during the PPE. Ended that right then and there.
Soundness. I can forgive a slightly off flexion but if the rads are not good, I never ever buy. I forgive some minor imperfections and this is where my vet’s advice comes in. I also walk away from a horse that is particularly back sore (I get rads on the back/neck if I am paying over $5k or so). I typically palpate the back and if there is spine-related soreness, I do not go any further, unless I have already fallen in love with the horse. In that case, I x-ray.
Strong preferences.
I look close at the canter and gallop but a huge gallop is not a requirement. But I aim toward horses that are not necessarily 4* potential prospects. Most of mine are more suited for 2*. I forgive gaits. I have also found some very crappy movers who could jump the moon, and some awesome movers that were stupid over fences.
Big sweeping walk. Trot with suspension.
Nice eye. I do not buy into the belief that big doe eyes are kind eyes. It is not the eye itself, it is the expression and where the eye goes.
Uphill, with a neck in front of me. This eliminates about 50% of all thoroughbreds.
Presence. This is difficult to explain but I typically like a horse immediately, if I am going to like them later. It is the overall picture, how it all ties together, and the feel I get from looking at and touching the horse. This is so difficult to explain, but those who look at, evaluate, and buy horses will definitely know what I am talking about here. I have looked at 10+ horses in a row that I knew I would not buy after only a few minutes of looking at and touching them. And I typically tell the seller before proceeding, out of courtesy to them and out of my desire to not waste time/energy.
Big plus
Curious, a bit spooky but not balky. I like horses that are a bit looky, who see something strange and perk up and need a good look-see. I do not like one that tries to run away from scary objects. Looky horses are careful horses, and careful horses are good jumpers. I also have had horses that had no fear of anything and they were okay jumping but not very crisp.
I adore thoroughbreds and have a prospect that I adore. That being said, I would also look at pedigree and at horses purpose bred for eventing. Thoroughbreds are terrific but for every great one, there are 30 that are not. You have to sift through a lot to find a good one. We have a lot of OTTB’s at the top of our sport. If you look at the pool of purpose-bred versus the pool of OTTBs, you will see that there are WAY MORE in the latter than in the former.
Be careful of sellers who are overly confident and boastful. No one can state that a horse is going to be a top event horse before it all happens. They may play with probabilities; but certainty is suspect.
There is so so much more. What fun to be shopping for a good prospect. I hate shopping for stuff but I LOVE shopping for horses.