Continuity mistake: When Frasier is in his apartment talking to Niles, Marty, and Daphne about choosing between Cassandra and Fay, the milk on the breakfast table changes position several times - from behind the coffee pot and near Daphne to the other side of the coffee pot in front of Martin.
Continuity mistake: Niles is trying to convince Frasier to play squash - he has his hands on the back of Frasier's chair. Cassandra comes out of the kitchen and we see one of Niles' hands on Frasier's shoulder. In the next shot, both hands are back on the chair, even though there was no time for this to happen naturally. (00:02:35)
Continuity mistake: When Frasier and Niles are in the cafe, Niles says "So what was Fay doing at your apartment last night?" When he says this, you can see he has just started to stir his coffee with a spoon. The shot then changes and the spoon has vanished.David Mercier