In "Goodnight Seattle", when Niles drops the ring, Eddie is eating from a plate. On a pulled-back shot, Eddie has licked the plate clean; but on the close-up of Eddie, the plate still has food on it. On the next away shot, Eddie is still licking a clean plate.
In Season Two's "The Friend", when Bob Reynolds wheels away from the table in Cafe Nervosa and out the door, he has the book on barbecuing on his lap. A split second later when Frasier exits behind him, the book is back on the table.
In "The High Crane Drifter" it's revealed that the apartment above Frasier's is a penthouse occupied by "Freddie Chainsaw," but in season nine the same residence is occupied by Cam Winston and the apartment is virtually identical to Frasier's and not a penthouse.
In the episode where Marty Crane is made President of the condo board over Frasier, Roz visits Niles and Daphne at their home. They offer her fruit salad. Towards the end of the scene, the amount and type of fruits in Roz's bowl changes between shots.
Throughout the scene where Martin brings out the new chair in "Bla-Z-Boy", and then Frasier brings out Martin's old chair, Daphne's legs keeps changing positions between shots. In some shots her legs are crossed, and in others, they are uncrossed. There also is a similar editing goof involving Daphne's legs in "Dark Victory".
The character of Frasier Crane is continued from his days in Boston on "Cheers," with his history pretty much established. However, even though in this series Frasier moves in with his father, on "Cheers" he made several references to both his parents being dead (i.e. "Cheers" episodes #8.21, #9.2).
The view of the Seattle skyline from Frasier's high-rise condo is an impossible view. The angle of the view, with the Space Needle in foreground and the skyscrapers in the background, is from north of downtown Seattle looking south. There are no high-rise condos north of the Space Needle, just one very massive hill, Queen Anne Hill. In order to get the view that he has, Frasier would need to have a ground-level house about halfway up the side of Queen Anne Hill.
The studio doors at KACL have handles that turn and click loudly when being opened and closed. A well-funded radio station like KACL would have padded door frames and doors with non-turning handles so that the doors don't make noise when opening and closing.
In the 2000 season premiere, Donny, a Seattle-based lawyer, launches a breach of contract lawsuit against Daphne Moon for backing out of their wedding. It is noted that the suit sought "$100,000 in punitive damages". Washington State does not allow for punitive damages in its courts. (Only cases heard in Federal Court are eligible for punitive damages in the State of Washington.)
As always with any long-running comedy series, there are some inconsistencies in characters' back-histories between episodes, but we generally excuse them. Discrepancies between this series and Cheers (from which it is a spin-off) are completely exempt from nitpicking (and some have been explained away, anyway).