Revealing mistake: At the end, in the shots of the kids dancing in class, the width of the wall between the shelf/coat hooks and the perpendicular wall with the door has bizarrely increased by 1-2 feet, but that extra width is gone in previous and following shots. Presumably they needed extra space for all the kids to dance so they moved that left wall.
Visible crew/equipment: After Barney tells Andy that the stranger may be a foreign spy Ed Sawyer walks into the courthouse, and behind him a crew member's hand can be seen catching the door just as the door closes.
Continuity mistake: When Otis begins his 24 hour sentence at Andy's house by staying in the guest room, Otis is wearing suspenders (braces) that are sewn together in the back in an X-shape, but when Otis leaves the guest room he's wearing entirely different style suspenders attached to his trousers.
Continuity mistake: When Opie and Arnold rush into the house and tell Andy and Aunt Bee about the party, in the shots facing the boys we can see the reflection in the mirror behind them, and there's a lamp atop the credenza, but in the shots facing Andy and Aunt Bee that lamp doesn't exist.
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.
Visible crew/equipment: While waiting for Barney to show up at the surprise party, Opie looks out the window and spots Barney, then when Aunt Bee and Thelma Lou take their places we can see both actors' marks on the floor - the chalk T-mark and the curved chalk mark.
Visible crew/equipment: After Ellie and Andy are turned away by Flint, when they walk back into the courthouse while Barney's lifting the chair, the chalk T-mark is visible on the floor.
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.
Deliberate mistake: After Barney buys the car from Mrs. Lesh, he and Andy are standing at the steps as he tells her not to look back at the car, and the next shot of Andy and Barney at the front door is the same shot from a few moments prior, but it's been flipped so they would be looking in the opposite direction. (00:09:30)
Continuity mistake: After Ellie tells the Taylors that she's withdrawing as candidate, when she and Andy head over to the rally at the hotel they pass the church, and in the long shot a large sign has suddenly appeared at the front of the church.
Visible crew/equipment: When 'FBI Agent' Jenkins congratulates Andy and Barney on behalf of J. Edgar Hoover they're all standing near the door, but when it cuts to Jenkins turning around and they all walk over to the desk, we can see the two curved chalk lines on the floor where Jenkins' feet were positioned, as well as the curved chalk marks on the upper step where the mayor was standing.
Andy and the Woman Speeder - S2-E3
Continuity mistake: At the end, when Andy and Barney are standing beside the woman's car her rear-view mirror has vanished in the closeups, but when she drives off in the long shot the mirror has reappeared.
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: When Opie's up on the ladder feeding the birds, in the overhead shot of Andy and Barney the mic's cable running up under Barney's pant leg is visible.
Aunt Bee the Crusader - S4-E15
Revealing mistake: Toward the end of the show the rooster, Bo, is drunk. If you look closely you can see the two strings pulling the rooster's neck sideways.
Revealing mistake: During the holdup at the bank, while the bank robber's holding Asa's broken gun, when Andy motions to Asa to duck out of the way in the closeup we can see the two thin wires to the left and right pulling the gun apart.
Barney Mends a Broken Heart - S3-E6
Visible crew/equipment: While Andy and Barney are with Daphne and Skippy at the Tip-Top Cafe in Mount Pilot, when Daphne's boyfriend walks in we can see the actor's chalk T-mark on the floor.
Mayberry Goes Hollywood - S1-E13
Visible crew/equipment: When Andy sees that several stores have made 'Hollywood' changes and then walks past the two men wearing suits, it cuts to Andy walking up to 'Hollywood funerals' and we can see the reflection of a crew member pushing the camera dolly with other crew members.
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)
Barney Hosts a Summit Meeting - S8-E21
Visible crew/equipment: When Andy and Barney start to get into their car, the boom mic is reflected in the front window. (00:10:10)
Answer: Those "boxes" are valences that used to be very common, before air conditioning. They allow for windows to be open during rain storms. They permit air circulation, without letting the rain in.