Plot hole: After the kidnappers kill the trooper and then one chases after the 2 eyewitnesses you can see that they are on a straight road (no turns and he catches up to them fairly quickly) but the next morning when Marge arrives on the scene of the 2 bodies she arrives from the direction of the trooper's body (the first shooting). After examining the bodies of the eyewitnesses she asks her deputy "where the trooper is" and the Deputy points in the direction she just came from and states "down the road a bit" how did she miss it? She would have had to have driven right past the trooper's crime scene to get to the 2 bodies.
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)