Continuity mistake: Abbie puts Niles' burning menu in the ice bucket where it remains for the rest of the scene, next to the bottle of champagne. Sometimes the menu is right up against the bottle and sometimes it is an inch or two apart - with no one touching either item.
Continuity mistake: Towards the end of the show, Frasier is explaining to Niles how he was stood up that evening; at the same time he is getting himself a glass of sherry. Niles says, "there was no emergency", we see Frasier repeat the actions to get a glass of sherry all over again.
Plot hole: Why can Niles hear what Frasier has said when he is inside the sensory deprivation tank? The whole point of these are that you cannot hear anything happening outside, so Frasier's ranting shouldn't have been heard inside. Similarly, why did the writers even have this scene with Frasier trying to talk to "Maris" when she was supposed to be inside it? Clearly they didn't quite know how soundproof they are.
Continuity mistake: During the opening credits, Frasier is playing the piano and Martin is going through papers on the table. When Marty says "I don't know, Eddie, must you?" there is something pink lying on the table. The shot goes back to Eddie and then back to the table, but the pink item has disappeared.
Plot hole: Niles makes up Frasier's couch to sleep in and when he lies down he lets out a yelp - and pulls a plastic pirate's hook from behind the cushions. Earlier Niles had appeared wearing a pirate's costume but at no point did he have a hook, nor did he go near the couch. His overcoat had been on the floor the whole time so it couldn't have come from there. It would have been difficult for it to get behind the cushions unless deliberately shoved there, which we never saw and makes no sense anyway.