Continuity mistake: Mrs Slocombe arrives with her hair in disarray, but once she walks away she only has time to put her coat away before she returns, by that time her hair is neat with no evidence of it being messy. She had to have returned quickly to hear Mr Humphries' story as she comments on it shortly after.
Continuity mistake: In the ruined photo taken of Old Mr. Grace and his nurse and secretary, the secretary is looking away from the camera and her leg is raised. However, in the scene preceding, she was looking at the camera and raised her leg after the flash went off. Also in the photo, both of Mr. Humphries' feet are in the air, though one was on the floor when the picture was taken.
Plot hole: When the robbers come onto the floor and go into Mr. Rumbold's office, the staff suggests calling the police, only to be told by Mr. Harman that the only outside line is in Rumbold's office. This, despite the fact that there have been numerous occasions throughout the series where characters made and received personal calls on the department phones (Mrs. Slocombe calling her cat, Mr. Humphries calling his mother, etc).
Other mistake: There is another counter on the floor, to the right of the stairs, that is usually staffed by two women besides Ms. Brahms and Mrs. Slocombe. But throughout the show when there is a on-floor meeting for the entire floor staff, these two are never included in the discussions.
Factual error: 2016 special - You Can't Teach An Old Dog New Tricks: Mr Harmon comments on a perfume called Belle, which he thinks is called "Belly", "I wonder what the next one will be called. Bingo Wings?" The episode is set in 1988 and the term "bingo wings" didn't exist before the 1990s.