The Taylors in Hollywood - S6-E8
Revealing mistake: While Andy, Aunt Bee, and Opie are at the Hollywood studio watching a scene being filmed, when Andy mistakenly stands up and interrupts the filming we can see the wall and shelves behind the director, and it should look familiar because it's the wall and shelves from Floyd's shop in Mayberry.
The Return of Barney Fife - S6-E17
Continuity mistake: At the reunion you see a sign that says class of '48. In season 3 They had a reunion and it said class of '45.
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)
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 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 Taylors in Hollywood - S6-E8
Visible crew/equipment: After the studio's security guard tells Andy they're filming on stage 40, when it cuts to the Taylors walking onto the set we can see two chalk T-marks on the floor (one in front of Andy's feet). (00:13:00)
The Cannon - S6-E11
Visible crew/equipment: After Warren accidentally fires The Cannon, in the following closeup of Andy commenting that he shot a station wagon, the shadow of the moving boom and mic is visible at the lower left corner of the screen. (00:22:05)
Revealing mistake: When Andy arrives at the studio for the publicity shots, while he's speaking to Mr. Michaels note that behind them we can see the Taylors' Mayberry kitchen cabinet/counter and a wall with their distinctive gold/silver design wallpaper. (00:03:50)
The Taylors in Hollywood - S6-E8
Revealing mistake: When Andy, Aunt Bee, and Opie are visiting the studio set while the scene between fake Aunt Bee and fake Andy is being filmed in a fake Taylor living room, note that behind the real Taylors we can see bits of the interior of their Mayberry house scattered in the background offstage. Some things to look for include their living room's stone wall with the mantle, their two reddish chairs (one with the fringe at the bottom), and a wall with their distinctive gold/silver design wallpaper. It's particularly funny when Opie looks at the "fake" living room set and comments, "Is that our house? It sure looks different!" while we the viewers can see the "real" Taylor set behind him. (00:20:35)
Other mistake: While Andy and Helen are rearranging everything in the squad car's trunk to make it all fit, Helen picks up the large thermos and written on its bottom we see "Reggie personal." Reggie Smith was property master on set. (00:12:45)
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.
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)
Visible crew/equipment: After Andy agrees with Helen to suggest a blind-date for Warren, when it cuts to Warren walking into the courthouse, we can see set lighting and equipment behind the door instead of the usual painted backdrop. (00:01:55)
Other mistake: During the scenes at Goober's filling station we can see a few brand names covered. For example, the "Chew Mail Pouch Tobacco" wall thermometer sign has its brand name "Mail Pouch" covered with black tape, and all the yellow "Pennzoil" cans have half the letters of their brand name painted yellow, so all the cans read "en zo."
Opie's Job - S6-E1
Visible crew/equipment: When Goober walks up to Andy outside the grocery/meat store, before Goober picks up the apples we can see a chalk mark on the ground between his feet.
Visible crew/equipment: When Floyd and the other men are with John Canfield joking about Floyd's barber skills, a chalk T-mark is visible on the ground by the man with his foot on the bench. Then when Canfield walks away we can see more chalk marks on the ground.
The Bazaar - S6-E5
Visible crew/equipment: After Andy shouts at Goober to get his pickup truck away from the courthouse, it cuts to Andy, Warren, and all the women in the courthouse, and we can see the chalk T-mark on the floor at Warren's feet.
The Return of Barney Fife - S6-E17
Continuity mistake: When Barney and Andy are at the Class Reunion dinner, there is a band playing, but the accordionist's fingers aren't in sync to the music.
Revealing mistake: When the bus pulls up for the Taylor family to leave for Hollywood you can see behind the bus a California freeway with cars moving in both directions. I don't think that Mayberry had a freeway right outside town. The scene is fixed when they return in "Aunt Bee on TV." There has been a tree put there so the road wouldn't show.
Visible crew/equipment: After Goober has deliberately removed the distributor cap from Andy's squad car, when Andy slams the hood shut we can see the reflection of the filming crew on Floyd's window. Then when Andy and Helen drive away in Goober's truck, the microphone pops up into view at the bottom of the screen. (00:17:20)
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.