The Legend of Barney Fife - S6-E18
Continuity mistake: At the start of the episode, while Barney's in Andy's kitchen he's wearing a shirt with large distinctive designs, but when he and Andy walk into the courthouse Barney's shirt has completely different designs (though it's the same type of shirt). (00:01:00 - 00:04:00)
The Legend of Barney Fife - S6-E18
Revealing mistake: When Andy and Goober enter the Taylor house Andy calls for Aunt Bee, and just as she walks out of the kitchen we can see the end of the set wall inside the kitchen. (00:14:20)
Continuity mistake: Aunt Bee won a garbage disposal in "Aunt Bee on TV" and it's one of the gifts she keeps, but she buys another one in a later episode, "Lost and Found," with insurance money.
Wyatt Earp Rides Again - S6-E20
Continuity mistake: After Opie tells Andy that he knows he'll win at high noon, it cuts to Warren in front of Floyd's shop (with its furniture conspicuously missing), but later when Andy walks over to Floyd's shop its facade has changed. Note differences such as window structure, and the name "Floyd's Barber Shop" painted on the window which wasn't in the earlier shot. (00:18:05 - 00:24:15)
Aunt Bee Learns to Drive - S6-E21
Continuity mistake: The car which Aunt Bee buys from Goober has a rearview mirror which repeatedly disappears and reappears between shots, throughout the episode.
Aunt Bee Learns to Drive - S6-E21
Other mistake: Aunt Bee says that Andy had to drive her to Clara's and Andy says it's not a bother, but Clara has always lived right next door.
The Gypsies - S6-E23
Continuity mistake: While Andy and the townsfolk are at the courthouse during the storm, when The Gypsies walk in the two men are wearing wet rain ponchos and the woman is wearing a striped serape, but then she's wearing a wet raincoat, and it switches yet again in following shots. (00:21:45)
The Gypsies - S6-E23
Factual error: The Gypsies manage to operate a shortwave radio without an antenna or AC power.
Continuity mistake: When Andy and Floyd stop by Wally's station, Flora's wearing a striped uniform shirt and there are a bunch of cars lined up waiting for service. Then later, when Goober gets back from vacation and shows up at the station, in the first shot of the station Flora's wearing the striped shirt and all the cars waiting for service are the same as the earlier shot, but in the closeups Flora's now wearing a solid color uniform shirt. (00:06:05 - 00:09:20)
Continuity mistake: During the scenes at Wally's station, the two prop gas pumps each have just one fuel dispensing hose/nozzle which switch between being between the two pumps, or with their hoses/nozzles on the opposite side of the gas pumps.
Other mistake: The end credits list actor Burt Mustin as playing a character named "Jud", but after Howard's introduction to everyone in The Lodge, when Andy and Howard are saying goodbye to all the members, Andy says, "See ya, Burt" to actor Burt Mustin. Then at the start of the voting scene even the subtitles read "Burt" instead of "Jud." (00:09:15 - 00:12:55)
The Barbershop Quartet - S7-E3
Other mistake: The end credits list actor Burt Mustin as playing a character named "Jud", but while Andy and the rest are waiting for Howard to show up for practice, Andy, Wally, and Aunt Bee all use Burt Mustin's real name "Burt" instead of his character's name, "Jud." Then during the rest of the episode Andy still calls him "Burt" a few more times. (00:02:00)
The Barbershop Quartet - S7-E3
Other mistake: At the end of the show Andy talks about the upcoming bowling league. Howard shows Andy his injured left thumb. This should not affect his bowling since Howard is right handed.
Continuity mistake: During the baseball game, when Opie slides into homeplate we can see the concrete mixer truck in the background beside the building, but when Helen shows Aunt Bee the photo she took of Opie the truck is gone. (00:09:40 - 00:23:25)
Other mistake: Helen is sitting on the first base side, there is a play at the plate and Opie is called out at the plate. Near the end of the show, Helen has a picture she took of the play and it show Opie to be safe. The picture angle is taken almost directly from home plate and at a low angle. The backstop is also blocked half the way up. She would have had to be on the field behind home plate for such a picture. It's also very close up but the camera she is using doesn't have any zoom feature.
Aunt Bee's Crowning Glory - S7-E5
Visible crew/equipment: When Andy removes the mirror from the wall (for Aunt Bee), the reflection of a studio light is visible in the mirror just as Andy starts moving it. (00:18:40)
Visible crew/equipment: When all the Darlings walk out of Andy's kitchen with a bag full of Aunt Bee's muffins, the actor's tape mark is visible on the floor, where one of the Darling boys looks down and stands. (00:20:25)
Revealing mistake: While Goober is up in the tree with the fake owl, when Andy and the Darlings leave the house they are standing on the porch, and after seeing the 'second omen' the Darlings leave the porch and head to their pickup truck, but what's funny is that we can see their reflections in the window behind Andy as they all walk off camera, turn around and just stand there, and we can still see those actors' reflections in the shots facing Andy and Aunt Bee even as the Darlings supposedly drive off.
Visible crew/equipment: After the squad car is hit from behind Andy rushes over to Goober, and a T-mark (it's a bit darker than the color of the dirt road) becomes visible on the ground (just as the woman gets out of her car) at the bottom of the screen, right behind Andy's feet. (00:01:25)
Visible crew/equipment: Just as Aunt Bee and Opie pull up in front of Wally's station, the reflections of crew and equipment are visible on the surface of the car. (00:06:00)
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.