Visible crew/equipment: At the start of the last scene, when Andy and Rogers walk out of the courthouse the reflection of a crew member is visible on the surface of the car, and as Rogers drives away we can see the reflection of the camera dolly on the car, and also the shadow of the boom mic on the ground.
Visible crew/equipment: When Rogers shows Andy the statistical chart and they get a call about the domestic violence, the two chalk lines (for feet position) can be seen on the floor below the bulletin board to its left, which is precisely where both Rogers and Barney have stood at different times during this scene.
Revealing mistake: When Barney pulls the jail cell window out of the exterior brick wall, as it's being yanked out we can see that the brick wall is only thin drywall during the stunt, as well as the fact that the wall around the breakage differs significantly between the closeup and wide shots.
Continuity mistake: When Otis begins his 24 hour sentence at Andy's house by staying in the guest room, Otis is wearing suspenders (braces) that are sewn together in the back in an X-shape, but when Otis leaves the guest room he's wearing entirely different style suspenders attached to his trousers.
Visible crew/equipment: When 'FBI Agent' Jenkins congratulates Andy and Barney on behalf of J. Edgar Hoover they're all standing near the door, but when it cuts to Jenkins turning around and they all walk over to the desk, we can see the two curved chalk lines on the floor where Jenkins' feet were positioned, as well as the curved chalk marks on the upper step where the mayor was standing.
Other mistake: Aunt Bee refers to losing the pickle contest 10 years in a row (which Clara corrects to 11, leading us to believe they have been in the same competition), but this is her 2nd year in Mayberry, having not seen Andy for years in her season 1 intro as the "new housekeeper", (or Opie - ever) in season 1.
Visible crew/equipment: After Andy and Barney walk out of Floyd's shop, when Andy makes a crack about Barney's ears being uneven Andy walks offscreen, then Barney shouts at him as Andy's presumably already down the block, but in their reflection in the window behind them we can see that as soon as Andy walks offscreen in this closeup, he actually stops and leans over where we see the camera's located.
Character mistake: In the beginning where Karen is introduced to Andy and Barney in the coffee shop, her name is Karen Worth, then at the last where she is called as winner of the Skeet Shoot her name changes to Karen Moore.
Other mistake: When Andy, Barney, and Opie switch Aunt Bee's kerosene tasting pickles with the store bought pickles during 'operation pickle switch', they put the store pickles in Aunt Bee's mason jars with silver lids and Aunt Bee's pickles in the store's jars with labels and white lids. Their plan is to give away the store jars with Bee's pickles inside, and we see Barney placing the last of the store jars into his bag, but when Barney stops the driver to give him his safe driving award, we see Barney giving away the mason jars with silver lids even though it's the store jars he should be getting rid of.
Other mistake: After Barney gets annoyed over Andy's lack of confidence in his ability to be a sheriff, Barney gets on the phone with Sarah and asks to be connected to the mayor of Greendale, then it cuts to an exterior shot and the parked car has plate number "DC 269," the same as other vehicles in some earlier episodes.
Other mistake: Mr. Wheeler's license plate number is 'DC 269' which has been seen in earlier episodes. Then at the end, when the locks-keys man pulls his truck into the driveway, it has plate number 'VT 772' also seen in episode 2x5, when the pretty woman drives into town.
Continuity mistake: On the curb outside the courthouse there's a fire hydrant beside the pole, which not only changes its position on the sidewalk in different scenes, but it has also vanished at the end when Rogers drives away, as Barney crosses the street.