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: In the episode "Call Me Irresponsible" from season 1, Frasier and Catherine are making out on Martin's old chair, which starts vibrating (probably because they inadvertently hit a switch on the side). But in "Give Him the Chair", Frasier tries to replace Martin's old chair by buying a new one which has a built-in massager. When Martin tries it, he looks horrified and says "That's disgusting", even though his beloved chair had the same feature.
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: About two thirds of the way through the show, Marty and Frasier are talking in the living room. Marty is sitting in his chair with a glass of milk. One shot shows him with both hands on his milk, and then he puts his left hand on the arm of the chair. In the next shot his hands are around his glass of milk again.
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).