The shield emblem on the staff car's left front fender changes from early in the show to the end, from being a blue/black cross on a white background to a white dagger on a blue/black cross..
In 'The Captain's Car' a Rolls Royce was donated to Mainwaring as his staff car, although it's possible it was taken back by the donor who was just awaiting petrol.
When Pike and Sylvia break down on the way home, it is supposedly late at night. However, it is clearly sunny. British summer, however, means it can be light well beyond the time Pike and Sylvia were supposedly stranded.
When Pike orders another raspberry drink he is surprised at the price. As he'd already had one he would know what the price was.
When Hodges introduces his niece in the office, there are several boom microphone shadow movements over Sgt. Wilson's face and head.
Jean Gilpin didn't have a licence and it shows: she tries (and fails) to start the car, announces that it starts well, tries to start it again and runs the starter motor far too long. It still didn't start.