Continuity mistake: Daphne is on the phone with her mother and has a dust rag in her hand. She holds out the phone towards Frasier and when the shot goes back to her, the rag is all bunched up in her hand - it has been loose before. There was no time for her to change it.
Continuity mistake: Frasier, Martin, Niles, Daphne, and Diane are having dinner at Frasier's and he gets up to get dessert. Niles offers to help him and puts his table napkin in his chair. As he walks towards the kitchen, though, the napkin is lying next to his plate.
Character mistake: In the scene where Jerome Belasco turns up unannounced at Frasier's apartment, Daphne tells him she's psychic and guesses that he's an osteopath. After Jerome leaves, Daphne says she thought he seemed very nice. Frasier turns to her and says, "This from the psychic who said he was a chiropractor."
Continuity mistake: Marty gives Niles a mug which says "World's Greatest Psychiatrist." He puts it down on the coffee table with the lettering facing the camera. He then leaves with Daphne and a few shots later, we can see the mug again, but it is turned around and we see the blank side. When Frasier leaves the apartment, it's back in its original position. Nobody ever touched it.
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.