Continuity mistake: As Barney is turning in his deputy equipment to resign his handcuffs are hanging out of his pocket. After Barney states he wishes someone would just commit a crime, Opie enters. As the camera pans out so Andy can ask Opie what happened, the handcuffs are no longer in Barney's pocket, and he didn't put them on the desk with the rest of the equipment. (03:03:00 - 03:11:00)
Andy's Trip to Raleigh - S8-E4
Visible crew/equipment: When Andy gets back from Raleigh and shuts the car door, just as the camera moves in closer we can see the reflections of two crew members (the one wearing a yellowish shirt looks as if he's pushing the camera dolly), on the window behind Andy.
Continuity mistake: When Andy finds the old newspaper clipping in Helen's box he tucks it into his pocket, but when he's looking at the clipping while he's at the office the newsprint on the back is entirely different. Then when Howard reads it the newsprint changes yet again.
Aunt Bee's Big Moment - S8-E23
Continuity mistake: When Opie asks his father how to spell the word "Renaissance," Aunt Bee holds a plate of assorted cookies, but when she stands up and heads to the kitchen the type of cookies on her plate have changed.
Ernest T. Bass Joins the Army - S4-E3
Visible crew/equipment: When the two men are carrying the window pane past Barney and Andy, we can see the reflection of a few crew members' legs.
Visible crew/equipment: After Andy and Barney have asked Otis if he's seen the billy goat, when it cuts to Opie on his bike we can see the two white tape marks on the ground where Andy and Barney stand.
Barney and the Governor - S3-E15
Deliberate mistake: In the closeup of the license plate on the governor's car, the words over and under the plate number are deliberately taped to be unrecognizable. The original words are "drive safely" and "North Carolina."
The Pageant - S5-E11
Continuity mistake: While Andy and Opie are sitting on the couch, Andy's on the phone talking about having to tell Aunt Bee to drop out of the play, and there are two candlesticks with candles on the server behind Andy, but in Andy's closeups they're gone.
Three Wishes for Opie - S5-E14
Other mistake: When Andy asks Floyd and Judd about who he's supposedly marrying it cuts to Helen walking down the street, and the grocery store has only one display window, when it should have two windows, and there's a brick wall adjacent to the window which shouldn't be there. (Shot at the soundstage not backlot).
Revealing mistake: While Barney's explaining to Andy his new interest in realty they both walk to the back room, and just as they pass through the doorway the camera tracks too far to the left, so we can see the edge of the set wall and the rest of the back wall is missing. (The following shot is from the direction of that missing wall).
Barney Runs for Sheriff - S5-E21
Other mistake: When Floyd tells Barney that he doesn't want to put Barney's campaign poster in the barbershop window because it spoils the decor, the reflection we see in the mirror behind Floyd is the church's front door and steps. This makes no sense at all because the reflection should be whatever is to the right of Floyd's, but the church is the opposite way - to the left, past the courthouse. The view in the mirror is a painted backdrop, but it's still quite wrong.
The Arrest of the Fun Girls - S5-E28
Continuity mistake: When Helen and Thelma Lou walk into the courthouse and find Andy and Barney dancing with Daphne and Skippy, there are lipstick marks on Barney's cheek, but when Andy gives Skippy her sweater and radio then tells them both to go, in the closeups the lipstick marks are gone, though they reappear momentarily.
Opie Flunks Arithmetic - S5-E30
Continuity mistake: At the end, when Helen shows up at the courthouse and tells Andy that Opie got a B+ on the test, her purse vanishes and then reappears while she's sitting on the banister beside the desk.
Opie Flunks Arithmetic - S5-E30
Continuity mistake: When Barney, Andy, and Aunt Bee are at the table after Opie goes upstairs to study, in shots that face all three of them there are things on the middle shelf to the left and right of the center plate, but then in other shots either the left item is gone, or the right item is gone.
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.
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.
Continuity mistake: Barney tries showing Andy what the "part bloodhound" dog can do for The Manhunt so Barney blows a dog whistle, and in the closeups of the growling dog the rope that was tied to his collar is suddenly lying on the floor, but it's back on his collar in the following shot.
Continuity mistake: When Barney, Thelma Lou, and Andy are driving to Andy's blind date they're all in the squad car. When they're parked in front of Helen's house in the closeup shots the light mounted on the roof has vanished, the glass windshield's gone, the car's interior differs, etc., and then it changes back. The roof is black, then white, as well.
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.