Continuity mistake: When Daphne is playing darts with Frasier, he says, "I scored again," and we see her sweater sleeve is pushed up her arm. But when she comes back with, "That's not something we hear very often," the sleeve is completely covering her arm. In the next shot, it's pushed up her arm again. There was no time for the adjustment.
Continuity mistake: Roz is at Frasier's to tape some CDs, and we see her take some cassettes out of her purse. The shot goes to Frasier and then back to her - the cassettes have disappeared. After he explains that he can't stay, we see Roz open her purse and take out the cassettes again.
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.