Continuity mistake: When Marty is writing the letter to Doc before he goes into the future, the word "disaster" is the only word in the last line. Yet when Doc shows Marty the letter when it has been all taped up, "terrible disaster" is the last line. Also, the handwritring in each letter is different (i.e. Marty's signature). They are two different letters.
Continuity mistake: When Marty is being chased by the Libyans, a few times it cuts away to show the instrumentation in the car. The first time the odometer reads 33064.2 and the trip counter says 88.8; the second time it's goes back a little to 33061.8 and 86.4. A few seconds later we see it as he turns the corner, and it goes down again from 33,061.8 to 32,994.4, the trip counter has gone from 86.4 to 19. In fact it's a completely different set of instrumentation - the bottom of the needle's angled in some shots and flat in others.
Continuity mistake: When Doc and Marty are at the Twin Pines Mall and Doc is telling Marty about the car etc, he says "never mind that now, never mind that now..." and Marty has the camera facing the ground, yet later when Doc in 1955 is watching the video, you can see him saying that on the video.
Continuity mistake: When Marty is standing over his father's bed in 1955 persuading him to date Lorraine, there is a hair-dryer in Marty's belt that seems to disappear between different frames (note the hair dryer's cord leading to Marty's right hand, as he holds it behind his back). Really obvious. [This scene was originally longer, including a part where Marty removed the hair dryer from his belt. Still a mistake, but there's the explanation.]
Visible crew/equipment: When Doc Brown loses his footing on the clock tower as it begins to toll, in the next shot, when he swings about towards the clock the thin black stunt cable attached to his waist, that leads around the neck of the statue, is visible. Do not confuse it with the thick silver electric cable Doc is holding.
Continuity mistake: In the cafe in 1955 Marty goes into the phone booth to look up Doc's name in the phone book. When he finds it the name is on the left hand page, but when he tears it out of the book he tears out the right hand page. One could say he turned the page to tear it right-handed, however when he is asking the waiter if he knew where the address was, he is reading the address on the same side he tore out - check the tears on the edge.
Continuity mistake: When Marty chases after Biff on the borrowed 'skateboard', Marty is wearing a dark grey belt and a red/blue print shirt under his red/beige jacket. However, when Marty is hanging on to the front of Biff's car as they turn a corner (and in another shot), Marty (stunt double) is wearing a light brown belt and solid tan shirt.
Continuity mistake: At the end of the movie, when Doc Brown leaves Marty at his house, he then backs up the DeLorean, turns the car around and passes in front of Marty doing 35 mph at the most, but just as he leaves the frame you can hear the swishing noises the car makes when it travels through time. No way could Doc have reached 88mph.
Factual error: When Biff is chasing Marty (on the 'skateboard') around the Town Square, you can see curb cuts at the corners. Aside from a trial program in Kalamazoo, Michigan in the '40s, they didn't spread further until well after 1955.
Factual error: The electric guitar Marty plays at the "Enchantment Under the Sea Dance" is a Gibson ES345. This guitar model debuted between 1957 and 1958 yet he's supposedly playing it in 1955. It would have been more accurate to have him using a Fender Telecaster (1950) or Stratocaster (1954) or a Gibson Les Paul (1952).
Continuity mistake: Marty has a brainwave about how to generate the power needed to get back to 1985, and shows Doc the flyer he was given, which Jennifer had written "I love you" on the back of, with her number. Problem is that after Doc grabs the flyer, we can see the back of it, and it's totally blank.
Continuity mistake: In 1955, when Marty drives the DeLorean to the 'start here' line that Doc painted on the street, the line and large white lettering are faded and filthy, but when Marty gets out of the car to insert the hook pole, the line and lettering are bright and immaculate. When he finally starts the DeLorean they're back to faded and filthy.
Continuity mistake: In the cafeteria, when Mr. Strickland is glaring at Marty and Biff, the table Marty was sitting at with George is behind him. When Biff leaves there is a close-up of the table and both the lunch tray and Marty's jacket have changed positions significantly from the previous shots (George had already left).
Continuity mistake: When Marty is in 1955, he is in shock and walking on the pavement when a car comes by. In the first shot when it comes towards him it has two speakers, on the second shot when it's moving away, it only has one. In the second shot of the car with only one speaker, the A-frame sign is also turned the other way.
Continuity mistake: When Marty is standing over his father's bed in 1955, Marty puts the headphones on George and then puts the tape in the cassette player. If you notice the hairdryer is on the left side of the belt and is touching his left arm, but when they cut to Marty holding the tape player out in front of him the hairdryer is on the right side.
Continuity mistake: In 1955, when Marty drives the DeLorean to the large 'start here' line that Doc painted on the street, in the first shot its front is up to the line completely covering the word 'here', with some distance to the last 't' in 'start'. Its position changes in following shots, such as when Marty tries to start the stalled car in the exterior shot, the DeLorean's front end is further back with the word 'here' perfectly visible.