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.
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.
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: At the end when Aunt Bea enters the courthouse, she doesn't close the door completely. You can see a hand from outside grab the door, and close it very slowly. (00:23:20)
Visible crew/equipment: When Wilbur's in his car he thanks Andy and the rest of the men for all the shoe sales. As he drives off, the reflection of filming equipment is visible on the surface of the car, and when the car is offscreen multiple chalk marks are visible on the sidewalk, at the feet of the actors.
Visible crew/equipment: When the men have the misconception that Wilbur is a TV show talent scout, while they're at the barber shop talking about it, just as Floyd moves his legs to get up from the barber chair we can see the white T-mark at Jason's feet.
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.
Audio problem: After the mayor tells Andy that his three daughters will be in The Beauty Contest, Andy phones Ellie and when she hangs up on him we hear Andy say, "Hello? Hello?" but we see him only mouthing the second hello.
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.
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.
Andy and the Gentleman Crook - S1-E21
Character mistake: When Barney and Gentleman Dan play gin, Barney deals only 8 cards. In gin, the proper amount would be 10.
Andy and Opie, Housekeepers - S1-E23
Continuity mistake: When Aunt Bee leaves the house holding her suitcase, in the interior shot the suitcase lid faces out as Bee exits the front door, but in the next exterior shot the suitcase now faces the opposite way (note the rivets) as Bee continues to walk out the front door. (00:04:05)
Andy and Opie, Housekeepers - S1-E23
Continuity mistake: Andy is sitting on the couch, next to him on the end table is a glass and what appears to be a dark bottle like a beer bottle. They cut away, when they return to Andy on the couch the glass and the dark bottle have disappeared.
Visible crew/equipment: After Barney tells Andy he overheard Dr. Benson and Ellie "talking marriage" while spying on them, Andy heads over to the drugstore. Then, when Barney enters the jail cell, we can see a crew member's fingers (at the lower right edge of the bars, behind the wall) push the cell door closed. (00:19:45)
Continuity mistake: About 20 minutes in, Barney knocks the announcement bell off the door trying to sneak out of the pharmacy, Andy comes in one minute later and the bell rings when he opens the door. You can see the bell swinging above the door.
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.
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.