When Ed and Peggy are talking outside the shop, Ed's position changes with different camera angles.
When Lou Solverson is speaking with the Blumquists, the red lamp on the sewing machine table switched sides from right to left. It can be seen behind Lou Solverson.
At the start of the program a movie narrator describes the invasion of South Korea by North Korea, how US Marines would push the North Koreans back, and how Truman offered unlimited military aid to South Korea. The screen then says that the place and time is Fargo, North Dakota, 1951. BUT the North invaded the South in 1950, NOT 1951, and the US got involved in the fighting immediately.
Surround speakers can be seen on the wall of the movie theatre in the flashback to 1951. Stereo was years away, let alone surround. It really wasn't until the mid to late 1970s that surround sound became widespread.
At 12:45 on the wall in the Waffle Hut an IKEA wall clock PUGG is visible. IKEA opened its first store in the United States only in 1985, while the action takes place in 1979.
Hanzee finds a piece of broken headlight in the road, which he then conveniently matches up with Ed and Peggy's car. But the police previously spoke about finding broken glass in the street, and they would've taken said glass as evidence. It's extremely unlikely they would've missed a piece as large as the one Hanzee found, but even if they had, roads get plowed when it snows. It was snowing the night of the murder and the road had clearly been plowed at least once in the days between the crime and Hanzee finding the glass. It's simply not plausible for that piece of headlight to still be lying in the road so many days after the hit and run.