Plot hole: The two kidnappers are pulled over by the cop they eventually murder because they are not displaying temporary tags on their car. However, when Marge Gunderson shows up to investigate the crime scene the next morning, she figures out from the cop's citation book that the car had dealer plates (DLR.) If the car had dealer plates, they wouldn't need to display temporary tags, as tags are to be used when a license plate is not yet available for a car. There was no need to pull the kidnappers over, which makes the turning point of the story completely pointless.
Suggested correction: Temporary tags have a non-unique number and the designation of the dealer on them, the cop put down DLR and was then going to put the dealer's number when they produced the temporary tag which they said they had forgotten to place in the window. On a temporary tag, the "tag number" is either the date the temporary tag expires or one of a limited number assigned to that dealer so it is not unusual to see duplicate temporary tags on different cars on the same day - you would need the dealer's number which is rather small at the bottom of the paper to actually identify the car.
But the point is that the police officer said he pulled them over because they were displaying no tag. If there had been a dealer tag on the car, then why would the officer have pulled them over? That's what a dealer tag is for: so a car can be driven displaying it. Yet everything that happens after that is predicated on the officer having begun to write out the number *on their tag*.
Suggested correction: They probably just had the generic plastic dealership logo plate on. Not an actual dealer plate with a number, but something with a logo and "Gustafson Motors" on it (or whatever the name of the car lot was).
That means they have no tags and the cop wouldn't write DLR. A logo with the dealer's name isn't a dealer tag.
Continuity mistake: When Shep strangles Carl with the leather belt, as he is putting the belt around Carl's neck Carl's hands are caught inside the belt as the shot changes; but when the shot changes to a different angle his hands are seen outside the belt.
Continuity mistake: When the cop is shot in the head, no blood splatters onto Steve Buscemi's face, but there's blood on it in the very next shot.
Factual error: Per the opening credits, the movie is set in 1987, but when Carl leaves the airport parking garage, the billboard in front of him shows Northwest's "Bowling Shoe" livery which was not introduced until 1989.
Continuity mistake: When Steve Buscemi is shot in the cheek, he touches the wound, and in the first shot there's no blood on his hand, but in the very next shot there is.
Factual error: As a local Sheriff, Marge wouldn't be running an investigation into the shooting of a state trooper. She wouldn't have anything to do with it beyond responding to the initial scene. State investigators would investigate the death of a trooper, and since that event leads to the rest of it, the other events as well.
Character mistake: When Jerry and Scotty are talking in Scotty's bedroom, after his Mom's kidnapping, there is a poster behind Scotty's door. The poster says "Accordian King" but should have been spelled, "Accordion King."
Other mistake: When Jerry returns home to find that Jean has been kidnapped, he sees the shower curtain rod that's loaded with rings, and there are more rings on the floor. There's way too many rings for a single shower curtain that would use 12 or so at the very most.
Continuity mistake: When Marge goes back to Lundegaard's office for the second time, he is working on writing the vin numbers on a paper on a clip board, scratching them off and scratching his pencil on a "I love golf" notepad. When he leaves his office to complete a "Lot Count", Marge is looking around Lundegaard's office. The camera pans back to his desk and the paper that Lundegaard was writing the vehicles vins on the clipboard is no longer there and has been replaced with another piece of paper/application.
