We both had the same sort of courage at our work, and John had oftener to hold us in than to urge us forward […]. (5.23)
Beauty recognizes Ginger as a kindred spirit because he can tell she puts heart and courage into what she does. Even though she's a little prickly at first, her bravery at having suffered such a difficult childhood impresses Beauty and they become close friends.
I dared not go forward, and I made a dead stop. "Go on, Beauty," said my master, and he gave me a touch with the whip, but I dared not stir. He gave me a sharp cut; I jumped, but I dared not go forward. (12.9)
This is probably Beauty's bravest moment. Even though his strongest belief is to obey his master at all times, he actually disobeys when he senses danger. His bravery saves the lives of Squire Gordon and John Manly, and they actually understand what Beauty's done to save them. Here Beauty shows that he's far more than just a servant to his humans—he's a companion and a courageous friend.
John said to me, "Now, Beauty, do your best," And so I did. I wanted no whip nor spur, and for two miles I galloped as fast as I could lay my feet to the ground. (18.6)
This is another incident that shows Beauty's true courage: He runs flat-out to save Lady Anne when her horse runs away with her, even though he doesn't know Anne particularly well. All that matters to Beauty is that a human is in danger; he doesn't even need to be told to sprint. Throw a flapping cape on him and he'd officially achieve superhero status.
Fearful as it was, no one stopped, no one turned back. Every moment the ranks were thinned, but as our comrades fell, we closed in to keep them together; and instead of being shaken or staggered in our pace, our gallop became faster as we neared the cannon, all clouded in white smoke, while the red fire flashed through it. (34.12)
Beauty's friend Captain is an incredibly brave horse who tells the story of his war experiences to Beauty. His descriptions of the battlefield are nerve-wracking and it's easy to imagine that this would have been a very frightening experience for a horse. Captain does what his master asks, though, charging into battle despite incredibly scary circumstances. It's definitely a courageous act.
I had such perfect trust in him that while he was guiding me, I was ready to charge up to the very cannon's mouth. (34.6)
Captain explains the reason he was able to charge into battle: He trusted his master completely. Their bond gave Captain all the bravery he needed to make it through the fight. Aw.
[…] I did my best, as I always had done, in spite of cruelty and injustice. (47.15)
Even when Beauty's completely beaten down by a cruel master, he still does his very best—and this is perhaps one of his bravest moments. He doesn't give up at a time when it would be much easier to only do a half-hearted job, and he maintains his strong values throughout.