Deliberate mistake: During scenes at Walker's Drugstore there are many magazines by the wall near the door, which have tape/paint hiding part or all of their cover titles. Examples are: all the Sports Illustrated have tape across the word "Illustrated", the House and Garden has its "House &" covered, and Time magazine's entire title is blacked out.
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.
Continuity mistake: In the initial episode, and Runaway Kid in season 1, Barney is a cousin of Andy's. In subsequent episodes, they are unrelated.
Factual error: The badges they wear don't match the badges on their patches. If there's a badge on the patch, it's supposed to match their actual badge.
Other mistake: In some episodes Barney can sing beautifully, but in others he can't sing at all.
Continuity mistake: Scenes shot at Andy's porch have significant differences depending on the filming location, as there are two sets of Andy's porch. One is just the facade of the house in the 'town of Mayberry' in the studio's 40 Acre backlot; and the second is located in the studio's soundstage. Note in some shots the door's sidelights have vertical muntin bars, the screen door has a white handle, whereas in other shots the sidelights' muntin bars are gone and the screen door handle is black (screen door's gone season 5). Also, note the significant difference in the width of the wall between the parlor windows and the door, the height of the parlor windows from the floor, etc.
Other mistake: The radio communications consoles on the side desk in the courthouse have a few pieces of black tape covering their brand name (though we know it's Motorola from the visible "M" logo).
Revealing mistake: On most of the corners of the buildings, you can see seam lines running down between the different sides of the buildings. The bricks on the opposite sides usually don't line up with each other either even though they are supposed to be the same brick.
Continuity mistake: The fuel pumps at Wally's gas station (regular and super) switch sides between numerous episodes.
Factual error: A town like Mayberry with a mayor and council would have its own police force, not just a sheriff.
Other mistake: At the beginning of the series, Hope Summers' character is called Bertha Edwards. Later in the series she is Clara Johnson; still later she is Clara Edwards.
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).
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.
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 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.
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: 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)
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.
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.