Continuity: At the awards ceremony Frasier is being introduced to his rival Fletcher's mother. At one point Fletcher is holding his mother's hand but in the next shot their hands are nowhere near each other.
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Quotes
[Martin sees Frasier has a lot of cuts on his face from shaving.]
Martin: I thought you were just going to slit your wrists. Looks like you went for death by a thousand cuts.
Frasier: I cut myself shaving because I was shaving without water. Why was there no water? Because I had to move your chair, which gouged the floor, which made me call for Joe, who found bad pipes, which called for Cecil, who ate the cat that killed the rat that LIVED IN THE HOUSE THAT FRASIER BUILT!!
Trivia
When Bulldog calls up Frasier and asks him if he can sing a bit, Frasier stars singing the Gilbert and Sullivan song "Three Little Maids from School" which of course leads to great embarrassment. In the episode "Cape Fear" from The Simpsons, Kelsey Grammar guest stars as Sideshow Bob and also has bad luck with Gilbert and Sullivan music. Bob gets badly injured after the family sings the Three Little Maids song. Later he is asked to sing the score from the HMS Pinafore and it foils his plan. See more...
Frasier (1993) - 5 mistakes in "And the Whimper is..."
starring David Hyde Pierce, Jane Leeves, John Mahoney, Kelsey Grammer, Peri Gilpin (add more)
Updates: 
Series 10
And the Whimper is... (season 1, episode 18)
Continuity: When Roz and Frasier are sitting on the couch discussing how the other award nominees are courting the judges, the amount of champagne in Frasier's glass goes down dramatically without him drinking any.
Continuity: Early in the show, at Cafe Nervosa, Roz comes in and Bebe asks her to get her some coffee. Roz returns with a paper bag which she puts down on Bebe's right. In the next shot Bebe stands up with it in her left hand, but there was not enough time for it to change from the right to the left naturally.
Continuity: Just after Roz puts Bebe's bag on the table at the Cafe, we see her back at the counter - there was not enough time for this to happen naturally.
Continuity: Early in the show, Frasier is sitting in Cafe Nervosa with Niles and Bebe, drumming on the table with the fingers of his left hand. In the next shot his hand is pulled in, lying on the table in front of him.
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