In the episode Mum's Army (1970), Mrs. Fox reveals her first name to be Marcia, but in later episodes she is called Mildred.
Throughout the series LCPL Jones is the only member of the platoon to wear medal ribbons on his uniform. Frazer, Wilson, and Godfrey (and possibly others) all saw service in WWI and would have medals and ribbons, and would most likely have worn them proudly as well.
On multiple occasions throughout the series Bill Pertwee (Chief Warden Hodges) can clearly be seen mouthing dialogue of the other actors.
In the first episode, Sergeant Wilson mentions that he served as a Sergeant in the Royal Artillery during the First World War. But, in the last episode, he appears in his old uniform as a Captain.
Arthur Lowe frequently calls characters by the wrong names and then quickly corrects himself.
There is a running joke that Mainwaring cannot grasp the 24 hour clock. However, given that he is a high-ranking bank official, it seems unlikely that he could not handle a simple calculation.