Continuity mistake: Towards the end of the show, Daphne, Niles, Sherry, and Martin are in Frasier's bathroom where he is trying to take a bath. He starts talking to Sherry - she is sitting on the ledge at the end of the tub. From one shot to the next she has her left leg crossed over her right, then her right over her left, and then both legs uncrossed. She did not have time to shift her position and there is no "residual movement" that would have been natural had she been fidgeting while he talked to her.
Other mistake: When Frasier enters the apartment, Martin and Sherry are talking in the kitchen. If you listen carefully to what they're saying, their conversation makes no sense. Martin says something along the lines of "Daphne is really a very nice girl" before suddenly, in a very different tone of voice, he says "You're right, we're not getting enough privacy around here". Seems like a big chunk of conversation was edited out.