Continuity mistake: At the beginning of the show, Frasier is sitting with Martin and Niles in Cafe Nervosa telling them about the radio play. As Marty says "we were a simple people", Frasier manages to open his briefcase, take out the script, put it on the table, and put his briefcase back on the floor and be sitting still - all in about 3 seconds. This just wouldn't be possible in real time.
Plot hole: During the scene when Frasier is in Cassandra's hotel room, he phones Roz and puts his phone back in his jacket pocket. He hangs his jacket on the back of the chair and eventually gets into bed with Cassandra, who soon falls asleep. He talks to her but she doesn't answer. We then hear him phoning Roz - but there was no time for him to get out of bed and get the phone - and no sounds of him doing this.
Continuity mistake: Niles and Frasier are at Marty's old apartment and Niles sits down on the couch. Eddie is sitting next to him, staring. During this scene, the shots of Eddie show him moving from fairly close to Niles, to about 10 inches away, and then back again. He has no time to change position naturally.
Continuity mistake: Frasier is in the kitchen and gets the sugar out to sprinkle over the profiteroles. As he starts shaking we get a wider shot and we see there is sugar all over the stove top, pastries, plate, and counter - way more than there should be for the amount of shaking going on.
Continuity mistake: At the beginning of the show, while Noel is getting Frasier to sign his Star Trek petition, Roz puts a blue plastic cassette into the machine to Frasier's left. During the scene it changes to a black cassette, without anyone touching it, and at the end of the scene Roz walks off with a blue cassette in her hand.
Revealing mistake: Near the end, in Cafe Nervosa, the guy that wants to sue Frasier gives Niles a push. Niles stumbles backwards, knocking over some chairs and the coat rack before falling down, taking the table by the window with him. If you look closely just before he hits the table, the woman sitting there (rather than moving out of the way), very subtly pulls on the table, so as to upend it so Niles can go down with it.