Visible crew/equipment: While Andy and Otis are talking about women Barney opens the door and rushes into the courthouse, and behind him we can see a crew member's hand (looks like a woman wearing bracelets) catching the door as Barney swings it shut.
Continuity mistake: After Frankie pays for the chewing tobacco in the drugstore, in the next closeup of Ellie the cigar box on the shelf behind her has black lines scribbled over all its identifying brand names (which is Van Dyck 43's), but when Frankie walks out the door, in Ellie's next semi-closeup that cigar box is unmarked again.
Visible crew/equipment: At the Scobey house, after Ben sits in the unstable chair and it breaks, when Andy and the Scobeys rush over to him we can see the two curved chalk actor's marks on the floor, where Ben had been standing moments before.
Continuity mistake: While Andy and Barney are trying to sell the town cannon, when they're talking to the shop owner who's seated in the rocking chair we can see the area between the squad car and storefront, with various items for sale, but in the next shot the covered cannon has suddenly appeared - and it's tall enough to have been in view in the previous shot had it actually have been there.
A Plaque for Mayberry - S1-E25
Visible crew/equipment: While everyone's waiting for Otis to arrive at the plaque presentation in Mayor Pike's office, after Andy admits to the mayor that he asked the real Otis to show up, it cuts to the mayor walking behind his desk, and we can see the chalk T-mark on the floor where Mrs. Bixby stands.
Runaway Kid - S1-E6
Visible crew/equipment: After Andy talks to George's father on the phone, he gets up and stands near the fireplace while explaining to Opie why he broke his promise and called George's father, and when Opie's on the couch asking if he can run off sometime, we can see the white tape mark on the rug where Andy was standing moments ago.
Continuity mistake: When Barney walks into the chicken coop there's a window on its door, but when Barney hides from Flint there's no such window on its door.
Visible crew/equipment: After Barney tells Andy that the stranger may be a foreign spy Ed Sawyer walks into the courthouse, and behind him a crew member's hand can be seen catching the door just as the door closes.
Other mistake: When Ralph Mason walks into Walker's Drugstore and asks to purchase a tin of pipe tobacco, the tobacco tin that Ellie puts on the counter has black tape across the top covering its brand name, which is "Tuxedo."
Visible crew/equipment: When Andy walks into the barber shop while Barney's complaining to Floyd about his sideburns, we can see the white T-mark on the floor at the bottom of the screen, as well as chalk marks where Andy's standing.
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.
Visible crew/equipment: While waiting for Barney to show up at the surprise party, Opie looks out the window and spots Barney, then when Aunt Bee and Thelma Lou take their places we can see both actors' marks on the floor - the chalk T-mark and the curved chalk mark.
Visible crew/equipment: After Opie tells Ellie everything that Andy told Aunt Bee, Ellie tells Opie that she's on the trail of a baboon, and when it cuts to the next shot of Ellie coming down the stairs we can see the chalk T-mark on the floor where she stands.
Visible crew/equipment: After Ellie and Andy are turned away by Flint, when they walk back into the courthouse while Barney's lifting the chair, the chalk T-mark is visible on the floor.
Continuity mistake: After Ellie tells the Taylors that she's withdrawing as candidate, when she and Andy head over to the rally at the hotel they pass the church, and in the long shot a large sign has suddenly appeared at the front of the church.
Visible crew/equipment: While Andy and Barney are standing beside Bobby Fleet's vehicle, Andy mentions that it's too bad they're all not staying overnight so they could hear Jim play, and when it cuts to the next shot we can see the reflection of the overhead boom pole and mic moving on the surface of the vehicle.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: When Andy sees that several stores have made 'Hollywood' changes and then walks past the two men wearing suits, it cuts to Andy walking up to 'Hollywood funerals' and we can see the reflection of a crew member pushing the camera dolly with other crew members.
Continuity mistake: After Eddie Brooke offers Barney five dollars and asks to be let go, it cuts to the State Police car passing the "drugs prescriptions" storefront, then Barney and Eddie trip and scuffle for two consecutive shots, and when it cuts back to the State Police car they haven't even passed that storefront yet. Additionally, that storefront is only two stores down from the coffee shop where Barney and Eddie are.
Visible crew/equipment: After Ellie tells Barney that she'll be going to the dance with him, when it cuts to Andy sitting on the porch we can see Aunt Bee's tape mark before she walks over.
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)
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.