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.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: Before the producer's arrival in his car, when Barney's telling Andy that it'll serve him right if they don't put him in the picture because Andy's not wearing his new uniform, we can see the moving shadow of the boom's hinge and mic on Andy's shirt.
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.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: Near the end, after the producer tells the townspeople to get back to normal and out of their trick or treat outfits, when the crowd disperses we can see some curved chalk lines on the ground near the producer's feet, to mark actors' positions.
Mayberry Goes Hollywood - S1-E13
Revealing mistake: When Andy is listening to the Hollywood producer talking about how he wants the town back the way it was before, over Andy's shoulder the "window" in one of the stores is blowing in the wind.
Mayberry Goes Hollywood - S1-E13
Continuity mistake: Barney's fancy new uniform has a badge on it when he first models it for Andy. During the ensuing festivities, it's missing.
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.