Corrected entry: In the beginning scene, when Doc Brown sends his dog one minute into the future, a minute later, they see the DeLorean again. But the reality is that they wouldn't, because the car would have also still been a minute ahead of them.
Continuity mistake: When Marty arrives in 1955, he puts the DeLorean behind the sign, and walks into Hill Valley. When Marty and the 1955 Doc come back later, the DeLorean is facing the other way round to what it was at the beginning (you can tell because at the start Marty pushes the BACK of the DeLorean AWAY from the Lyon's Estates billboard, but when they come later, the FRONT is facing TOWARDS the billboard). (00:35:05 - 00:51:15)
Continuity mistake: In the scenes where Marty enters the Lyon estates, if you look close you can see a stop sign, but if you watch the next two movies the stop sign has been removed. (00:11:50)
Suggested correction: I watched the films and the stop sign is still present.
Corrected entry: At the beginning of the movie, when Lorraine is telling her children about the dance, she turned to George and said it was the same night as the terrible storm. At the courthouse, the storm was fierce - wind, thunder, and lightning. At the dance, however, it was a pleasant evening. No evidence of a storm whatsoever.
Correction: "Same night" doesn't mean "at the exact same time". The dance and the storm both happened the same evening.
Correction: The school is some distance away from the courthouse, and the storm simply hadn't arrived there yet.
Corrected entry: When Lorraine follows Marty back to Doc's in 1955 and Doc goes to answer the door, he says to Marty (before opening it) that it's his mum. Lorraine must have heard them through the door yet she never questions it.
Correction: Two possible reasons she never asks about it. First, she simply did not hear him. You're assuming she did, but there's no evidence of it. Second, asking questions would be tantamount to admitting to eavesdropping. It's rude to eavesdrop and so Lorraine would hardly give herself away by asking questions about something she wasn't supposed to have heard in the first place, and which would have sounded nonsensical anyway.
Correction: Besides, even if she did hear him, so what? She knew she obviously wasn't "Calvin's" mum. She would have assumed his mum looked like her and Doc mistakenly thought that was who was at the door. But if she still decided to question it, I just gave an explanation that they would have used.
Corrected entry: Between scenes Marty's shoes change, e.g. in the skateboard chase scene he is wearing black converse shoes, but when he is in the time machine travelling back to 1985 he is wearing white Nike trainers.
Correction: Not a mistake. After Marty says "Yeah, I can spend a week in 1955." Doc says "first, we have to get you out of those clothes" - this is so Marty blended in with current fashions of the 50's. When Marty is going back to 1985, he claims that he isn't going back in his 50's clothes - "you think I'm going back in those clothes?.", or something to that effect. It was all done to make Marty less suspicious to the people around him.
Correction: That's not how it works, at least in this movie. Einstein jumped a minute ahead instantly, Doc and Marty "catch up" after a minute of their normal time. Einstein isn't permanently a minute ahead of everyone else on the planet forever.