You find out a horse's bravery by going to "explore" and then the first time they spook it will reveal their bravery rating. Same order as the regular breedings - World Class best, Elite, and so on.
Pearls - if you have a Round Pen, you can train up the Bravery. But you may want to first find the ones by exploring that are Elite or WC to train up, as it can get spendy.