Opie Steps Up in Class - S8-E5
Continuity mistake: When Aunt Bee drives up to the Hollander house with Opie, in the wideshot the car's rearview mirror is affixed to the windshield, but in the closeup the mirror has vanished.
Continuity mistake: At the start, Opie and his friend rush to the front door of the same house to try and sell their products - note the house next door (and the fact that it's three houses down from Andy). The kids then run across the street and have the door slammed in their faces, but then it cuts to the woman next door as she walks to her front door - note the house behind her is that first house Opie went to, even though all the boys just crossed the street a few moments ago.
Continuity mistake: While Barney and Thelma Lou are stalling Andy and Helen in the squad car, note the long whip antenna is missing (the empty bracket that always holds the antenna in place is over the rear passenger door) when they drive off to Andy's, but when they pull up to the house the antenna's on properly.
Continuity mistake: While the Darlings are at Andy's house the first time, on the hutch that's against the wall behind the sofa, there's a vase on the left side of the bottom shelf which vanishes and reappears.
If I Had a Quarter Million Dollars - S5-E22
Continuity mistake: After Barney makes his way around town spreading The Rumor that he has loads of money, when Barney walks into the courthouse he removes his jacket and hat then hangs them on the coat tree, and the hat keeps moving under and over the jacket between shots.
Continuity mistake: When Barney walks into the bank and turns on the light, we can see there's no damage to the side of the door, but when Andy walks in he points out the area where the door was jimmied.
Continuity mistake: When Barney's outside the bank loading his gun with the bullet, there are window blinds on the inside of the doors' windows - nothing else, but when Barney's inside the bank there are white shades between the glass panes and the blinds. As for the bank's large side window the opposite happens - there's a white shade between the pane of glass and blinds in the exterior shot, but in the interior shots we can see through the blinds' slats and there's no shade.
Opie's Job - S6-E1
Continuity mistake: We see the exterior of the grocery/meat store a few times during this episode, and there are differences in things such as the actual facade of the building, the view through the doorway, etc.
Opie's Job - S6-E1
Continuity mistake: When Opie and Billy are both heading to the grocery store to apply for the job, the church is behind the boys and they're standing in front of the 'new and used furniture' store - both of which are to the right of the courthouse (when facing the courthouse), and to the left of the courthouse are the barbershop and the grocery/meat store.When the boys run across the street, they're coming from the opposite direction.
The Bazaar - S6-E5
Continuity mistake: When Andy looks out the courthouse window and sees Goober's pickup truck filled with the ladies auxiliary group, Andy rushes out the front door and there's a wall right behind him where there shouldn't be, and although there's a bulletin board on the wall it still differs from the real one, plus the wall's air vent and wall trim are missing.
Continuity mistake: When Aunt Bee, Andy, Ollie, and Nora are in the kitchen for breakfast, the bag of flour vanishes from the counter depending on the angle of the shot.
Continuity mistake: After Andy convinces Barney to take off the uniform and wear his suit, when they pull up to the courthouse in the squad car the rear-view mirror is on properly, but in the shot facing the windshield the mirror has vanished, and then it reappears.
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.
Continuity mistake: When the mayor finally arrives for the band's marching performance he parks his white car in front of the wood bench where he'll be sitting in a moment. After the fake band's performance, when the mayor tells Andy that the band can go to the band festival, in the shot of the real band members cheering while Andy pats the mayor on the back and also when the Fleet Band drives by, we can see the entire street in front of the mayor and his car has vanished.
Andy's Rich Girlfriend - S3-E2
Continuity mistake: At the start, when both couples get in the squad car the rear-view mirror is gone in the closeup, but in other shots it's on properly.
Continuity mistake: Aunt Bee won a garbage disposal in "Aunt Bee on TV" and it's one of the gifts she keeps, but she buys another one in a later episode, "Lost and Found," with insurance money.
Continuity mistake: At the end, when Aunt Bee serves dessert her spoon and napkin are on the table, but in the next closeup of just Andy and Opie her spoon and napkin have vanished, then when Bee's seated she removes her napkin and her own dessert and spoon should still be on the table, but in the closeups of Andy and Opie they're not.
Continuity mistake: During the scenes at Wally's station, the two prop gas pumps each have just one fuel dispensing hose/nozzle which switch between being between the two pumps, or with their hoses/nozzles on the opposite side of the gas pumps.
Continuity mistake: On one occasion when Mr. Weaver leaves the courthouse to walk around the building to look in through the cell window, he goes in a counterclockwise direction. However, when the scene switches he is approaching from a clockwise direction.
A Plaque for Mayberry - S1-E25
Continuity mistake: About 2/5 through the show, Barney comes into the office with a family tree chart. The edge of the paper is on the bottom of the roll sticking out to the left, there's a cut to a different angle and the edge of the paper is on top, not sticking out and still facing to the left. On top, it should be facing to the right.
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.