Deliberate mistake: In the beginning of the movie, we see Marty play his black Ibanez during the band try-outs. Later, we see it on his bed when Doc calls him and when he wakes up the next day in this alternate timeline. We never see him bring it home after school or when he goes home at night before dinner.
Suggested correction: Maybe one of the band members brought it back to his house for him. Maybe he forgot it and picked it up again later, off camera. Maybe he has 2. Maybe the guitar he is using at the try-out isn't his. So many possibilities.
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. (01:07:00)
Continuity mistake: When Biff and his goon friends are in Biff's car, as they chase Marty on his borrowed 'skateboard', the car's rearview mirror repeatedly disappears and reappears, and the side mirror changes from round to square repeatedly. (01:06:50)
Other mistake: The car that almost runs Marty over has the vehicle registration plate of 6S 48405. Later on, when Marty is hit by Sam, Sam's car has the same plate. (00:35:49 - 00:42:02)
Continuity mistake: When we first see the machine in action, the travel is shown from three different angles. In the second, a line of fire passes directly between Doc's legs, but in the third angle, they pass to the left of both of his legs. (00:21:00)
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. (01:36:30 - 01:38:55)
Visible crew/equipment: When the clock tower ledge breaks, Doc Brown slips and loses his grip on the electric cable. In the next shot the stunt wire pulling the thick electric cable towards Doc's pants is visible. (01:37:50)
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. (00:29:20)
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. (01:40:55)
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. (00:36:30)
Continuity mistake: When Doc and Marty are at the Twin Pines Mall and Doc is telling Marty about the car, he says "never mind that now, never mind that now..." and Marty has the camera facing the ground, but later when Doc in 1955 is watching the video, you can see him saying that on the video. (00:18:50 - 00:50:05)
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.] (00:59:45)
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. (01:35:25)
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). (01:23:50)
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. (00:10:55 - 00:52:05)
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. (01:05:58)
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. (01:36:30)
Factual error: When Marty hits his head on the DeLorean's steering wheel, the car's horn honks. This is inaccurate to the design of real-world DeLoreans, as a DMC-12's horn button is actually located at the end of the turn signal lever rather than in the steering wheel. (01:39:27)
Suggested correction: You can't see his left hand and he could have accidentally pushed it when he hit his head. Or Dr. Bown customized the steering wheel so the horn is in it, like he customized the entire interior of the car.
I thought the point was that the car was so unreliable that when hit his head on the steering wheel, the horn went on. Makes sense to me. I own a DeLorean by the way.
Revealing mistake: 1955 skateboard chase scene around town square. Note in this scene a metal device has been added between the kicktail and the back truck of the skateboard to create the sparks, then disappears and reappears throughout the whole chase scene. (01:03:40)
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. (01:42:00)
