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.
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.
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.
Factual error: PC Angel's collar number is initially HF 777. Whilst a borough code of HF doesn't actually exist there would be a borough code for the Metropolitan Police as it is big enough to warrant one. I'll guess the HF to be for Hot Fuzz? Once PC Angel is promoted to sergeant and also transferred to Sandford his collar number changes to SF 777. Not only would the borough code not be required in the much smaller Sandford, but SF is a Metropolitan borough code for Finchley. The number would also change from 777. Three digit numbers are reserved for constables whilst sergeants would have either a one or two digit number, as in Martin Freeman's HF 90. The twin sergeants, Turner, both have collar number SF 101, when they should have different one or two digit numbers, likewise sergeant Fisher who has SF 137 and the later promoted sergeant Butterman, SF 128. It appears that the format of these collar codes is left to the discretion of each constabulary but the number of digits denoting rank seems to be uniformly accepted throughout the UK.
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.
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.