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.
Continuity mistake: At the beginning of the show, Niles, Frasier and Daphne are at Cafe Nervosa. Daphne is standing up talking to Eric the server and Niles gets agitated, and spills sugar everywhere. Eric throws down a dish towel so they can clean it up. The dish towel appears and disappears several times during the scene.
Continuity mistake: Early on in the show, Martin is trying to get Eddie to shake a towel off his head while Daphne stands behind the dog, folding laundry. In one shot she is standing with the folded towel over her arm; in the next shot she is folding it again. When she goes to the door, there is something red on top of the laundry basket, but when she returns the towel is back in the laundry basket (it's not the towel Marty put on Eddie's head: that's sitting on the coffee table).
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.
Revealing mistake: The view from Frasier's apartment shows the Space Needle prominently in the background, however, the elevator that takes visitors from the ground to the observation deck is frozen in time midway to the top. This was corrected after the first season.
Continuity mistake: At the beginning of the show, we see Marty add some beer to his champagne glass until it's almost full - just then Bulldog grabs his SeaBee award from the coffee table and now Martin's glass is almost empty, even though he's had no chance to even take a drink.