When Sam buys a newspaper the newspaper stand he has it in his hand then goes to the lift, when he gets out of the lift he has no newspaper.
When Sam is in the jail cell, looking at his host's reflection in the mirror Al appears behind him and Sam asks him, if he is Bogie, which Al replies No, Bogie is currently on Long Island with Audrey Hepburn filming Sabrina.
This is factually inaccurate as filming for Sabrina didn't begin until September 1953 and Finished in November of the same year.
The Film wasn't released until the following year. (1954)
Play it Again Seymour is set in April 1953,
7 months before filming actually began.
When Sam asks Alison where the tickets are to, her response "Does it matter?" is clearly an overdub. It is likely the original reply was "Casablanca" to fit with the rest of the ending, but which may have raised copyright issues. Not a goof because it is not an overdub. Casablanca is a city in Morocco, not just a movie title, so there would not be copyright issues.
Although this episode takes place in April 1953, the June 1956 issue of "Official Detective Stories" is on sale at Seymour's newsstand.
Obvious use of stunt double in the elevator shaft.