Continuity mistake: After Dan meets Gretta he walks her to the subway. A girl with a cell phone is coming up, but in the next shot she disappears.
Continuity mistake: Travers rides in a stretch limousine and Ralph, her driver, always puts her in the seat farthest back at the rear window. However, whenever she is seen inside the vehicle, she is sitting right behind Ralph, and easily reaches over and slides shut the window between the front seat and the back, even though there should be a wide space between where she is sitting and the driver's seat.
Continuity mistake: Near the beginning of the film, Llewyn goes into a subway station. The train that comes in is a different car model than the one he is sitting in in the next shot. Also, the exterior shot is of a train that did not yet exist in 1961. That model was introduced around 5-6 years later.