Continuity: When Frasier gets up to get out of the plane seat between the married couple, he is clutching his coat and boarding pass in his left hand/arm but when he reaches the aisle, they are in his right hand.
Great sites
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
During the show's eleven year run, there have been many celebrities who voiced the callers on Frasier's radio talk show, but are uncredited; most phoned in their lines, instead of recording at the studio. On the DVDs, the list of each season's celebrity phone voices are noted after the season finale. Some celebrities included: Mary Tyler Moore, as Marjorie, who couldn't confront her boss; Christopher Reeve, as Leonard, an agoraphobic who was afraid of wide open spaces; Matthew Broderick, as Mark, who liked to see himself on stores' video cameras; Elijah Wood, as Ethan, who got beaten up at school for being smart, and then insulted Frasier's answer; Ben Stiller, as Barry, who couldn't stop crying long enough to even talk about his problem; Ron Howard, as Stephen, who heard people talking to him through his radio; Carrie Fisher, as Phyllis, who had insomnia (Frasier fell asleep while on air, before he heard her problem) and Frasier advised, "Things will look better in the light of day, my advice is to sleep on it."; real-life husband/wife Phil Donahue and Marlo Thomas, as the married couple Larry and Sophie, who wanted a few of her girlfriends to come over on Super Bowl Sunday, but Larry refused unless Frasier was actually able to answer a football question on air; and many other enjoyable ones. See more...
Frasier (1993) - 5 mistakes in "Frasier's Imaginary Friend"
starring David Hyde Pierce, Jane Leeves, John Mahoney, Kelsey Grammer, Peri Gilpin (add more)
Updates: 
Series 10
Frasier's Imaginary Friend (season 5, episode 1)
Continuity: When Marty does his exercises, Eddie watches from the recliner and his bottom is just about in the center of the seat, but as Daphne heads to the door, his posterior is much closer to the arm of the chair, even though he never moved.
Continuity: When Frasier is talking to Niles and Martin in the restaurant (when he is supposed to be meeting Kelly) and the level in his wine glass alternates between being half full and almost empty and back again.
Continuity: Near the start of the show, Daphne is helping Marty with his exercises and as she goes to answer the door, the towel on his little table changes position noticeably.
Continuity: At the beginning of the show, Frasier is on the plane sitting next to a woman who finds out he's only on there because he wants to pursue her. She asks the flight attendant if she could change seats - but the shot before showed that the attendant had completely walked by their seat but when the woman asks her if she can move she has not passed their seat yet.
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