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.
Plot hole: At the garage, Shep says to Jerry that he does not know Carl, and doesn't vouch for Carl, only for Grimsrud. Obviously the two guys don't know each other. So, how can Carl react by shouting "Shep" when he is attacked in the bedroom by the Indian while humping the escort? (00:13:05 - 01:04:10)
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 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.