Visible crew/equipment: While Andy's in Opie's bedroom when Opie's thinking about running away, the moving shadow of the overhead boom mic is visible, first on the wall behind Opie's headboard and then on the door. Also note, when Andy shows Aunt Bee how to hold the baseball bat and moves her into position, we can see a long white tape mark on the grass (since it's possible that Andy placed the tape, however unlikely, it's being noted here).
Mayberry Goes Hollywood - S1-E13
Continuity mistake: When Andy's done showing Hollywood producer Mr. Harmon around town, he says, "Here we are, right back where we started, old oak tree," then mentions climbing that tree as a boy, and later says, "You're gonna cut down the oldest oak tree in the entire town?" But viewers have seen the entire area in front of the church (season 1 "Ellie for Council") and there isn't a large old oak tree. The fake tree was placed in this episode as a plot point between the producer and the townspeople.
Visible crew/equipment: After Aunt Bee plants the seeds of suspicion in Andy's mind about the shoe salesman, when Andy and Barney cross the street the reflection of the moving boom mic is visible on their squad's windshield, just as Andy walks off to the hotel.
Andy and the Gentleman Crook - S1-E21
Continuity mistake: After Gentleman Dan has taken Barney's gun, Aunt Bee, Opie, and Andy walk in on his attempted escape, and when Barney leaves the jail cell his badge inexplicably vanishes and reappears.
A Plaque for Mayberry - S1-E25
Visible crew/equipment: When Andy and Barney walk into Mayor Pike's office, they stand beside the two women from the Historical Society, and the faint chalk curved marks are visible on the floor where they stand.
Other mistake: Notice the license plate of Rogers' car is "DC 269," which is the same plate number as Andy's squad in a few season 1 episodes such as "Ellie Comes to Town" and "A Feud is a Feud."
Continuity mistake: On the curb outside the courthouse there's a fire hydrant beside the pole, which not only changes its position on the sidewalk in different scenes, but it has also vanished at the end when Rogers drives away, as Barney crosses the street.
Continuity mistake: While the squad's siren switch is stuck in the on position, Barney and Rogers get out of the car to open the hood, and suddenly there's a pencil sticking out of Barney's shirt pocket for Rogers to use to tap the condenser box, to stop the blaring siren.
Continuity mistake: On Flint's farm there's a cow tied to the back fence, which can be seen in exterior shots throughout the episode - when Ellie, Andy, and Barney are there. After Flint tells Barney to get off his land, in the exterior shot when Barney walks into the barn we see that cow tied to the back fence, but next shot inside the barn we see that very same cow in the stall - note the black/white markings.
Continuity mistake: While Barney's on the farm trying to get Frankie, twice he heads into the barn through the right-side entrance door within the tall sliding double barn doors (about 12 ft high), but when it cuts to the interior shots he's entering through the barn's left-side entrance door, plus the fact that it's not within huge double sliding doors. (This interior set is seen again in "Barney Gets His Man.").
Visible crew/equipment: When Aunt Bee and Opie leave the house and get in the car, the reflection of crew and filming equipment can be seen on the surface of the car.
Continuity mistake: At the "Hen Party", Jeff Pruitt says hello to Thelma Lou and she states she is going to get coffee, then goes to the kitchen. However, when Jeff walks across the room and looks back, Thelma Lou is back in the same area. She never walked back from the kitchen.
Factual error: When you walk into the courthouse the first jailhouse to the left has a window. But the courthouse is connected to another establishment on that side. A window would be impossible.
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.
Mayberry Goes Bankrupt - S2-E4
Continuity mistake: In establishing scenes throughout the first part of the episode, Andy is seen driving up to Meyers' home which seems to be located far from neighbors and on a dirt road. In the "renovation" scene and those following, Frank's home is located on a paved road with other homes nearby in the shot. Clearly a different set entirely.
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.
Andy and the Gentleman Crook - S1-E21
Visible crew/equipment: When Aunt Bee is in the kitchen preparing to take a tray of food to the prisoner she momentarily walks out of the kitchen through the swinging door. As she pushes the door open, a crew member's hand and arm can be seen behind the door.
A Black Day for Mayberry - S4-E7
Plot hole: There is no reason for a gold truck from Denver to Fort Knox to be going through Mayberry.
Opie Steps Up in Class - S8-E5
Continuity mistake: When Aunt Bee picks up Opie at Billy's house, she parks the car in front and walks up to the front door. In the wideshot there's a space of approximately 3-4 feet between the blue window shutter and the door trim under the pediment, but in Aunt Bee's closeup there is no space whatsoever.
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.