Visible crew/equipment: In the scene where BJ, Hawkeye, and Radar are in the motorcade talking about the horse, Radar is talking about how he doesn't want to give away the horse. He is standing in front of some barrels and when he moves, you can see some of the crew, the horse owner, and what looks like another horse in the background for 1 second.
Continuity mistake: When Colonel Potter's in his office finishing up his letter to Mildred, right before Margaret, Frank, and the others walk in, the things on his desk change their positions between shots - such as the paper tray moving from Potter's left to the right side of his desk.
Revealing mistake: In the shot where the horse kicks in the door, there's actually two shots - a kick, a quick cut, another kick. If you look closely, you can see how the hoof of one of the horse-leg props breaks off during the first kick. The second kick is performed by one leg only. If you look really closely - and probably use the slow-motion function - you can even make out the hand of the one wielding the leg through the hole made in the first kick. (00:18:10)