When Barney approaches Floyd's Barber Shop, the "Closed" sign is at eye level, just below the window's crossbar. In the close-up, the sign is at the bottom of the window. In the next shot, the sign is back at eye level.
Malloy uses Barney's gun to escape from the courthouse. In the next scene, however, Barney is back in uniform and has his gun in its holster. It's highly unlikely Malloy would have left the gun behind.
When Barney drives away with the trailer at the trailer park, it is clearly seen that the man driving the squad car is not Barney.
When Barney is talking to Opie, he's holding the dog on a rope. The rope slips out of Barney's hand when the dog starts to run off, but after Opie calls the dog back, the dog returns without the rope.
The crook's station wagon license plate has no state or year information.
When Barney is on the phone with the grocery store clerk who captured the barking dog, Sam, the sign to the right of the clerk says "Jam's 30 cents". The word jams should not have an apostrophe in it.
Barney asks Opie to go down to the barber shop to see if Floyd is there and tell him to come get Sam the dog. Yet, Floyd's Barber Shop is literally right next door to the courthouse. Barney and Opie are standing not more than ten feet or so from the shop. Therefore, Barney should be able to see whether or not Floyd is in his shop and wouldn't need to send Opie to check the shop.
When Barney says, "I thought you agreed to stay out of this case" outside the courthouse, his utterance doesn't match up with his lip movements. Shortly thereafter, when he says, "Come on, Sam", his lips aren't moving at all.
In one of the first scenes, when Inspector Horton is talking to Andy in his house, the camera angle is either too high or too wide, allowing the back wall of the set to be seen. If it was shot in a real house, the living room ceiling would be visible, but the top of the paneled wall ends and the uncovered wall studs are visible.
When Barney hauls Doc Malloy out of the trailer to arrest him at the end, he tells Malloy in reference to his gun, "No funny business! This baby is loaded with six bullets this time!" This is purely subjective but in retrospect, this seems like an odd statement for Barney to make. However, there may be an explanation for this line. When Doc Malloy is escaping jail and realizes Barney isn't another convict, but a cop in disguise, he grabs Barney's gun out of its holster on the rack and holds Barney at gunpoint. In the next scene, Andy finds Barney tied up and and releases him. Then when Andy and Barney are in Goss's Dry Cleaners and realize that Malloy's partner is a woman, Andy gives Barney six bullets to "load up". It's not clear how Barney got his pistol back since presumably, Malloy took the gun when he tied up Barney and escaped. It's possible that a scene was either written, or filmed and cut, in which Malloy realized Barney's gun was empty and abandoned the pistol. It would be the only explanation as to why Barney has his gun back and why Barney would make such a statement when arresting Malloy at the end.