Continuity mistake: When the sheriff goes to the hotel bar and talks to "Feathers", she picks up a bowl and then the camera angle changes and the bowl is back in the corner. Also, the shot glasses they use keep changing positions. They're a few inches apart then a foot apart, then close again.
Other mistake: When Joe shoots the person at the beginning of the movie the character is played by Bing Russell (Kurt's father). Later in the movie when Dude is ambushed at the entrance to the town, they give a guy his hat to impersonate Dude - it's Bing Russell again, after he was killed off at the beginning and why Joe is arrested in the first place.
Continuity mistake: The sheriff, Carlos and wife and Feathers are in the back room. The sheriff asks Feathers if she was sorry she missed the stage. When he does, Carlos' wife turns her head to look at Feathers then turns back to the right. Immediately after she does that, the scene switches angles and she is shown turning her head to the right again.