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.
Revealing mistake: During The Manhunt in the woods, Andy, Barney, the trees, etc., cast long shadows due to the overhead sun, but the problem is that we also see Andy's and Barney's shadows being cast in another direction as well, due to the set lighting on location.
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.
Continuity mistake: When Barney and Andy are looking at Barney's picture on the front page of the newspaper, first, the article has nothing to do with Barney or the arrest, and second, when Barney takes the paper from Andy he folds it with the front page on the outside, and the page we see is not the same front page from the closeup.
Visible crew/equipment: At the end, when Andy opens the door to Floyd's so Ed Sawyer can get a haircut, the set's large spotlight is reflected on the door's window.
Visible crew/equipment: While Floyd's napping in the barber chair, Andy's reading the newspaper, and when Barney walks in the actor's two curved chalk marks are visible on the floor, where he will stand.
Visible crew/equipment: While Barney's in the barber chair, Floyd starts talking about how he'd tried living at the back of the barber shop, and when the shot faces Barney we can see the reflection of a crew member's legs moving around - the only ones in the shop are Barney, Floyd, and the two men seated by the corner.
Continuity mistake: When Andy and Mrs. Silby are walking into the courthouse, in the exterior shot of Andy opening the door he's wearing the baseball mitt on his left hand, but when it cuts to the interior shot Andy's opening the door with his left hand and the mitt has vanished.
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.
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.").
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.
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.
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.
Plot hole: When Barney's telling Andy about the poem written on the bank wall, Barney blames Opie and doesn't mention anything about Opie using chalk, but when Opie walks in Andy tells him, "Barney says there's a poem written on the wall of the bank, and that you were standin' along beside it with a piece of chalk in your hand," and then in a few moments Opie explains that some big kids pushed the chalk into his hands. So it's impossible for Andy to have known that chalk was used to write the poem, and not a marker, a pen, or a pencil.
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.