Opie's Ill-Gotten Gain - S4-E8
Continuity mistake: Andy reads the inside cover of Barney's old schoolbook which has Barney's name "Bernard P. Fife" written out many times in it, but in two previous episodes Barney's middle name is either Oliver or Milton.
Continuity mistake: In this episode, during the scenes at Thelma Lou's house the front door is right beside a wall perpendicular to it, but in earlier and later episodes the front door is beside the dining room at the opposite side of the parlor.
Continuity mistake: While Aunt Bee and the party guests are gathered in the parlor, when Gomer hoots like an owl in the shot facing him note Floyd's eyeglasses - black with thick straight temples, but in the shot from behind Gomer and all following shots Floyd's wearing different eyeglasses - lighter color with curved tips.
Opie's Ill-Gotten Gain - S4-E8
Continuity mistake: When Opie's surprised by the bike, his first closeup is a flipped shot - note the bell's on the wrong side of the handlebar, etc.
Opie's Ill-Gotten Gain - S4-E8
Continuity mistake: When Barney is "reciting" the preamble to the Constitution, he messes up his hair. When they show him in close-ups it is messed up, but when it goes back to him and Andy in the scene it is combed. It does that a couple of times.
A Black Day for Mayberry - S4-E7
Continuity mistake: At the beginning of the show, two men walk into the courthouse. (As a sidenote, one of the men is Rance Howard, which is Ronnie Howard's real dad). Barney is the only one there, as Andy is out. The two men sit down to wait for Andy. Rance Howard is not wearing a hat, and he is in a light-colored suit. When Opie walks through the door looking for Andy, you can see that Rance is not sitting there any more, and now it's a man with a dark suit and wearing a hat. (00:01:55)
Continuity mistake: At the start, when Andy stops at Wally's station Gomer is telling Jason about his Trout bait, then someone pulls up behind Jason's car and they're bumper to bumper in the wideshot, but in the exterior shot facing Jason there's no car behind him, but that car is back again before he pulls away.
Continuity mistake: Otis is in the jail and can't sleep. He ask Andy for some water. Andy gets up, unlocks the door and then closes it again (clearly leaving the key in the lock). He comes back a few seconds later and the key is gone and back on the wall. (00:05:05)
Continuity mistake: At the beginning of the show, Barney and Andy are sitting inside a 1964 Ford patrol car. When they are shown up close inside, you can tell it is not the likeness of the same car. The back glass windows look more like a 1960 model possibly. It is definitely not a 1964.
Continuity mistake: When Andy's talking to Gomer through the window while Gomer's holding the pail, in the shots facing Andy the pail's handle is facing out, but in shots facing Gomer the handle is facing the opposite way.
Continuity mistake: When Opie shows Andy his arm with the chore list written on it, Andy uses his right hand to hold up Opie's arm, but in the closeup it's Andy's left hand holding up Opie's arm. (00:03:50)
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: In earlier episodes, we are told that Gomer can't do mechanic work. He just pumps gas and air: "There's no charge for air; air is free; we don't make no charge for air." But in this episode, Gomer has become an ace master mechanic.
Continuity mistake: When Otis makes up a poem about jumping rope, Barney lets go of the rope and Andy is left holding the other end while the rope is left stretched across the floor. One second later when Opie and his friend come running in, Andy is now holding the rope in his other hand all rolled up.
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.