Continuity mistake: In the closeup of Barney's yearbook picture his name is listed as "Bernard Milton Fife," and Andy even reads it aloud, but in the previous episode from season 1, "A Plaque for Mayberry" Andy says that Barney's middle name is Oliver.
Floyd, the Gay Deceiver - S3-E9
Continuity mistake: At the start, when Floyd and Andy walk into the barbershop, in the exterior shot Floyd's white barber jacket is on a hanger hanging from the coat tree that has a finial at the top of its pole, but in the next interior shot that jacket is hanging directly on the coat tree's hook, the hanger has vanished, and that finial has disappeared as well.
Continuity mistake: At the end, when Andy walks in on Barney showing Gomer how to draw his gun, Gomer draws his gun out of the holster which sends it flying behind him, and in the closeup the bookcase glass shatters, but in the wideshot when they all turn to look the glass is still perfectly intact.
Continuity mistake: When Gomer drops the gun from the roof it lands on the ground directly in front of Andy and Barney, who are standing on the sidewalk beside the courthouse, but the area where they're standing looks different between the overhead shots and closeups, and their shadows change as well. Additionally, the gun lands with its handle to the left, but in the wideshot the gun is lying the opposite way, with its handle to the right.
Continuity mistake: While Peggy's waiting in front of the drugstore she's splashed by a car driving through a puddle, and when she shows up at Andy's house the stains on her dress are different.
Continuity mistake: After Andy and Floyd talk outside the grocery Floyd gets into his car with the convict, and when they drive away we see all the food ads taped to both of the store windows while Andy's on the sidewalk, but when it cuts to Andy's closeups the taped ads have changed.
Aunt Bee's Medicine Man - S3-E24
Continuity mistake: Andy and Barney are playing checkers near the end of the episode. Andy can be seen making 2 moves in a row, skipping Barney's move.
Continuity mistake: When the mayor finally arrives for the band's marching performance he parks his white car in front of the wood bench where he'll be sitting in a moment. After the fake band's performance, when the mayor tells Andy that the band can go to the band festival, in the shot of the real band members cheering while Andy pats the mayor on the back and also when the Fleet Band drives by, we can see the entire street in front of the mayor and his car has vanished.
Andy's Rich Girlfriend - S3-E2
Continuity mistake: At the start, when both couples get in the squad car the rear-view mirror is gone in the closeup, but in other shots it's on properly.
The Darlings Are Coming - S3-E25
Continuity mistake: When Briscoe gets into his truck, his hat is wet from dipping it (5x) in a horse trough. The camera cuts away to Andy and immediately back to Briscoe, whose hat is now dry. Also, "she's always hood for 11 hat fulls," yet Briscoe only puts 5 hats of water in his engine.
Continuity mistake: In this episode, Ernest T writes Charlene a love letter. In a later episode when Ernest T goes back to school, it is revealed that he can neither read nor write.
Aunt Bee's Medicine Man - S3-E24
Continuity mistake: The pillow on the right side of the couch is there while Opie is talking to the Colonel. When Andy walks around after Opie leaves, it has suddenly disappeared. (00:16:00 - 00:18:00)
Answer: It was a joke. They were supposed to be dumb hillbillies.
Both. According to later interviews with both Andy and the Dillards band (the real name of the band that played the parts of the Darling children, of which Charlene (Maggie Peterson - Mancuso was also a member) the reason the Darling boys never had any lines was because they would've had to have been paid if they spoke. In return for appearing without lines or pay, Andy promised to get as much of their music onto the show as possible which, he did. This obviously gave them a lot of exposure.