Revealing mistake: On most of the corners of the buildings, you can see seam lines running down between the different sides of the buildings. The bricks on the opposite sides usually don't line up with each other either even though they are supposed to be the same brick.
Mayberry Goes Hollywood - S1-E13
Revealing mistake: When Andy is listening to the Hollywood producer talking about how he wants the town back the way it was before, over Andy's shoulder the "window" in one of the stores is blowing in the wind.
Revealing mistake: During The Manhunt in the woods, Andy, Barney, the trees, etc., cast long shadows due to the overhead sun, but the problem is that we also see Andy's and Barney's shadows being cast in another direction as well, due to the set lighting on location.
Revealing mistake: When Barney and Andy are searching the woods for an escaped con, they stop to rest and you can see the light level of the woods go up and down as the crew tries to adjust the reflector screen for the desired lighting.
Revealing mistake: When Andy tells Opie not to drop by the courthouse anymore after school, in Andy's closeups the shots are flipped - note the part in Andy's hair, the top of his badge, and his backwards buttoned shirt.
Revealing mistake: The woman in the foreground, as the siren is stuck, is carrying a paper grocery sack from "Hollywood Ranch Market" which was located on Vine St and Fountain Ave in Hollywood, Los Angeles. (00:09:35)
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.
Andy and Opie - Bachelors - S3-E4
Revealing mistake: After Andy and Opie choose between chicken and pot roast using a nickle, when Opie gives the nickle back to Andy the window over the sink is missing - the fake window wasn't put up on the set wall for this shot, but when the food burns the window and curtains are up there as usual.
Revealing mistake: During the holdup at the bank, while the bank robber's holding Asa's broken gun, when Andy motions to Asa to duck out of the way in the closeup we can see the two thin wires to the left and right pulling the gun apart.
Revealing mistake: In the scene where Andy starts reading Opie a story, watch when the first woman rings the doorbell. As soon as Andy opens the door you can see that the set all of a sudden gets a lot brighter, as if they turned more lights on.
Revealing mistake: The jailed criminals take the key and open the jail cell door. The problem is the dead bolt never moves with the key, showing the door was never locked to begin with.
Revealing mistake: Before Barney jumps out from behind the crate to surprise the two cops in the alley you can hear the director yell out "Don".
Ernest T. Bass Joins the Army - S4-E3
Revealing mistake: When Barney picks Andy up and they get back to the courthouse, they find Ernest T out of the jail cell again and breaking more windows, and in the shot of Ernest T leaping in the air we see it's actually a flipped shot - note the backwards "snack bar" in the window (that store is later when Ernest T tries to break the new glass for the courthouse).
Aunt Bee the Crusader - S4-E15
Revealing mistake: Toward the end of the show the rooster, Bo, is drunk. If you look closely you can see the two strings pulling the rooster's neck sideways.
My Fair Ernest T. Bass - S4-E17
Revealing mistake: At the start, when Ernest T. throws the rock at Mrs. Wiley's window and it breaks, the area around the hole flutters in the breeze - it's obviously not a glass window used for the stunt.
Revealing mistake: Barney and Mr Weaver scuffle in the dark. When Andy enters the store he turns on the light, but the light switch is still in the off position.
Revealing mistake: When Andy, Barney, Gomer, and the kids are at the campsite in the woods, during the daytime scenes the only shadows that should be seen are the ones being cast by the sun, but there are shadows being cast in different directions due to the set lighting.
Revealing mistake: At the end, in the shots of the kids dancing in class, the width of the wall between the shelf/coat hooks and the perpendicular wall with the door has bizarrely increased by 1-2 feet, but that extra width is gone in previous and following shots. Presumably they needed extra space for all the kids to dance so they moved that left wall.
Revealing mistake: When Barney blows the dog whistle to call Blue from the back room, the dog comes running and knocks Barney on his back, and we can see a bit of the cushion that's under Barney's back - it's visible within the shadow under him.
Three Wishes for Opie - S5-E14
Revealing mistake: Barney, convinced he can foretell the future, is in the darkened jailhouse, seated between Goober and Floyd at a table. Goober strikes a match and lights a lamp which is directly in front of Barney. The immediate illumination obviously comes from a studio light because the men's shadows on the wall are to the far left of the men, instead of directly behind them.
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.