Deliberate mistake: About halfway through the show, Roz and Daphne are in the kitchen talking. Daphne pours water into the teapot - but there is no steam coming from the kettle, nor any sound of a boiling/just boiled kettle. No self-respecting Brit would make tea with water that wasn't boiling.
Continuity mistake: Niles is filming is father in his recliner, trying to record his life story for posterity. Marty fakes a heart attack or death - and lies back in his chair with his left arm by his side. In the next shot his arm is crooked up around his head, but with no time for him to have moved it.
Continuity mistake: Near the end of the show, Roz and Frasier are at the radio station, in his booth. There is a clipboard lying on the console to the left. Roz goes to her booth (without the clipboard) and Bulldog bursts through the door on the left. The clipboard has disappeared. Shortly afterwards Roz goes back into Frasier's booth and the clipboard is back where it was.