Continuity mistake: Frasier and Roz are in Cafe Nervosa and Roz is looking at a newspaper while Frasier talks about giving Donny a bachelor party. Roz puts the paper in her lap, but in the next shot over her shoulder we can see that she is still holding the paper up.
Continuity mistake: Frasier's third date, Leslie, shows up at his apartment wearing a long off-white coat. He hangs it up beside the fireplace; for the rest of the scene we sometimes see the coat hanging with the left sleeve straight and sometimes with the left sleeve all crinkled up.
Revealing mistake: During the scene where Frasier is dropping Charlotte off at the train station (the first time) you can see the background behind him shows that the car they are in is turning left. Yet we can see his hands on the top of the steering wheel and during this time he never turns the steering wheel.
Plot hole: When Frasier is trying to get to the bottom of Niles' problem viewing Daphne in a real way, we see them both against past scenes in the show; Frasier watches and comments on some of them as he talks but because he is actually only discussing these events with Niles, he shouldn't be able to see a visual of anything such as the dress Daphne is wearing during the dance scene.