In Episode 6 of Season 1, Frasier telephones the police to report being ripped off by an art dealer. As a prank, Martin instructs Frasier to ask for "the fine art forgery department". Thus, Frasier asks for "the fine art forgery department". The police laugh at Frasier and disregard his call. Martin then tells Frasier that the police are too busy solving murders and robberies to investigate art forgery. However, art forgery is a crime, and the Federal Bureau of Investigation has an Art Crime Division. Thus, the police would have taken Frasier's call seriously and would not have laughed it off.
A new sump pump wouldn't cost anywhere near as much as a vacation to Italy. Fresh installation of one might, but the caller specifically says it's a "new" sump pump.
When Frasier (Kelsey Grammer) pondered how to get even with the art gallery that sold him the forged painting, he considered talking about his experience with the art gallery on his radio show. Niles (David Hyde Pierce) cautioned him that doing so would be slander, and the gallery would sue Frasier for everything he owned. Thing is, a truthful statement is not slander in the eyes of the law. However, Niles' point is that the gallery owner could still claim it was slander and then Frasier would have to endure the headache of a court battle even if he's ultimately victorious.
At the beginning of the cocktail party the pianist's hands doesn't sync with the sound being heard.
In the scene where Frazier and his father are calling the police, the actress who plays Daphne (Jane Leeves) can be seen standing just off set behind a wall waiting for her cue to enter the stage. She is seen moving around behind the wall.
In the opening scene, Frasier (Kelsey Grammer) begins to sing: "When the moon hits your eye, like a big pizza pie..." He is pretty poor at it and it gets people to call in. Unfortunately, Frasier cuts off right at the beginning of the song (as above) to take his first call, which Roz (Peri Gilpin) has miraculously been able to ID already. However, KACL is on a seven-second broadcast delay (as we learn in "Call Me Irresponsible"), so even if Roz had not taken the time to screen the caller for name and dilemma, the man would have had to call in before Frasier began to sing.