Continuity mistake: Near the end of the show, when Niles, Frasier, and Martin are at Duke's, Niles says, "I'm not here just for you, Dad" and he puts his hand on Marty's shoulder. The next shot shows him standing about two feet away, with his hands by his side.
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.