Continuity mistake: When Marty goes back from 1955 to 1885 and meets the Indians, note that none of the Indians shoots any arrows nor do any arrow hit the car. You can see that when he drives the car back into the cave. But when he gets back to the cave after the Cavalry ride, a long arrow sticks out on the side of the car.
Continuity mistake: When Doc's about to send Marty back to 1885 from 1955, and Marty's dressed up...when Doc's walking back and forth loading up the car - look at the shadow from the building. You see it changes by several feet in a few seconds as the scene was obviously shot throughout the day.
Continuity mistake: When Marty is inside the cave (where he hides the DeLorean), keep a close eye on the huge landscape and very large mountains behind Marty when he is screaming at the bear. The entire background changes when he runs out of the cave to just a few hundred yards of ground before it drops down, where Marty falls into the fence. What happened to the vast landscape and mountains?
Continuity mistake: When Doc has just consumed the 'wake up juice' and runs out of the bar and dunks his head in the horse trough, Marty and the bartender come out after him. You can see that Doc is now soaked and a trail of water is on the ground behind him from a previous take of Marty dragging him back in the bar, which hasn't happened yet at this stage. Only really noticeable in the cropped version.
Continuity mistake: When Doc and Marty are chasing the train on horseback, Doc gets on the train first and his horse pulls forward. Then when the camera shows Marty on his horse, another horse can be seen in front of him. When Marty jumps on the train, his horse falls back. But in the next shot you can still see the horse as it moves off camera.
Continuity mistake: When the Doc enters the destination time into the time circuits when he is sending Marty to 1885 from 1955, the time/date etc. Go in as he's typing it. However, in the previous films and later in the 3rd film when traveling back to 1985, the enter button must be pressed before anything appears for the destination time.
Continuity mistake: In the scene where Marty is driving the black Toyota pickup at the end of the movie, the truck's registration plate is missing - it can be seen when the truck is in the garage at the start of the film. Also the driver's door quarter light (vent window) is open. When we see the inside of the door it is closed.
Continuity mistake: In the end scene where Doc's train comes back to 1985, Marty and Jennifer are literally blown away by the re-emergence of the train. While they are flying in mid-air, the front of the train is passing them by and gets just about to the cab (where the ELB is). However in the next cut, Marty and Jennifer are on the ground watching as the train just starts passing them again, and the train comes to rest at a point before the previous "blown away" cut scene.