Factual error: At the end of the film, SGT Angel gets into his Police car Subaru Impreza, and on the bonnet it has the writing 'Sandford Constabulary'. This is incorrect, it would in fact be 'Gloucestershire Constabulary' as established at the start of the film that Sandford is in Gloucestershire. A Constabulary is always named after the County, never after a tiny village within that County.
Revealing mistake: When the Sergeant has the shotguns in the backpack on his back, and you see a close up of his face, you can see that the shotguns do not have triggers. On plastic prop rifles, the trigger is usually molded into the trigger guard in the pulled back position, and you can see this on these prop shotguns.
Continuity mistake: When Michael attacks Nicholas in his hotel room, during this scene objects being broken or shoved (such as stuffed monkey, radio, lamp, picture) change their position, or the characteristics of the actual damage, change from shot to shot. Additionally, when Nicholas is slammed into the picture it begins to fall forward, but next shot it rests upright against the wall.
Continuity mistake: When Angel is chasing the shoplifter, the distance between Angel and the gold statue keeps changing in different shots.
Visible crew/equipment: In the scene where the police are storming the local Somerfield shop, after pushing the trolleys into the butchers, they are attacked by others of the shops staff who are throwing fruit. In one close up shot of the fruit being thrown if you look carefully at the pineapple as its in the air there is a white part on the bottom. This suggests that the fruit being thrown are actually plastic. Slow motion is useful to see this but not required.
Other mistake: When Martin Freeman punches in the numbers on the telephone to call Steve Coogan, the numbers he enters are not shown on the telephone's screen.
Other mistake: When Sgt. Angel meets Joyce Cooper at the hotel, she is making a crossword puzzle. She said "Fascist" and then read the definition, but that word is not 7 across as she says, but 14 down.
Visible crew/equipment: When Angel stands after being shot in his bulletproof vest, the entire crew is reflected in his sunglasses. Most obvious is the boom operator with his microphone.