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.
The Arrest of the Fun Girls - S5-E28
Continuity mistake: When Daphne and Skippy drop in on Andy and Barney at the courthouse and want to "play jail," Skippy carries the radio and her sweater into the jail cell with her, but when Skippy asks for her ukulele we see the radio and sweater are now on the side desk, under the gun case.
Continuity mistake: After Andy and Floyd talk outside the grocery Floyd gets into his car with the convict, and when they drive away we see all the food ads taped to both of the store windows while Andy's on the sidewalk, but when it cuts to Andy's closeups the taped ads have changed.
The Keeper of the Flame - S2-E14
Continuity mistake: When Andy holds Barney by the shoulders while congratulating him for an outstanding job of detective work, Andy holds his notebook in his left hand, but when it cuts to a different angle the notebook's suddenly in Andy's right hand.
Ernest T. Bass Joins the Army - S4-E3
Revealing mistake: When Barney picks Andy up and they get back to the courthouse, they find Ernest T out of the jail cell again and breaking more windows, and in the shot of Ernest T leaping in the air we see it's actually a flipped shot - note the backwards "snack bar" in the window (that store is later when Ernest T tries to break the new glass for the courthouse).
Opie's Job - S6-E1
Other mistake: In the scenes outside the grocery store with Mr. Doakes, there are a couple dozen cereal boxes in the display windows - Post Grape Nuts, Post Grape Nuts Flakes, Post Alpha Bits, etc., and all the boxes have part of their names covered, such as "Rap Ut." It's pretty funny considering that TAGS cast did commercials for Post Grape Nuts, and other Post foods.
Visible crew/equipment: After Goober has deliberately removed the distributor cap from Andy's squad car, when Andy slams the hood shut we can see the reflection of the filming crew on Floyd's window. Then when Andy and Helen drive away in Goober's truck, the microphone pops up into view at the bottom of the screen. (00:17:20)
Emmett's Brother-in-Law - S8-E18
Visible crew/equipment: When Ben accompanies Emmett on his first call to Andy, just as they both enter the courthouse, we can see a crew member's hand outside reach out to catch the door as it closes. (00:12:40)
Aunt Bee's Restaurant - S7-E21
Visible crew/equipment: After Goober orders the chow mein dinner at the restaurant, in the following wideshot of everyone at the table, the curved chalk actor's mark is visible on the floor between Andy's and Helen's chairs. (00:12:20)
Aunt Bee's Restaurant - S7-E21
Continuity mistake: While Jack is trying to help Aunt Bee understand her fear, in a few of the closeups of Aunt Bee and Andy (seated on the couch) note the fruit bowl has moved and the candlesticks have vanished from the server, the large figurine on the left side of the hutch has vanished, and the flowers from the table have also disappeared. Then everything is back in their place. (00:21:10)
Malcolm at the Crossroads - S6-E3
Continuity mistake: At the start, Andy walks down the street and stops at the church steps to watch Ernest T. being a crossing guard, but when it cuts to the shot of Ernest T. we see the church steps yet Andy's gone, then when it cuts back to Andy he's still standing by the church steps.
Visible crew/equipment: After the shot of Milly holding hands with Howard (holding a picnic basket), when it cuts to Goober sitting in front of the courthouse the camera angle is low, so the wood framework and lighting equipment at the top of the courthouse set is visible, just as Andy walks over to Goober. (00:05:30)
Continuity mistake: After Clara cancels lunch with Aunt Bee, when Bee walks over to the table there are two candle holders on the server and a large picture surrounded by four small pictures hanging over the server. However, when Opie runs to the table, the left candle holder and bottom left small picture are both gone in his closeups. (00:19:40)
A Visit to Barney Fife - S7-E18
Other mistake: After Andy and Barney hear on the radio that another supermarket has been robbed they head over there, and in the shots at the supermarket note that many of the products on the shelves have their brand names covered with tape or scribbled over. For example, boxes of "Mrs. Cubbison's" stuffing/dressing and "Sunshine Cheez-It" are covered with tape. (00:16:40)
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.
Visible crew/equipment: When Barney and Floyd are spying on Sam while he's getting groceries, we can see both of the grocery store's windows and its clear glass, but when Sam drives off the left display window has been covered with a dulling spray resulting in a matte surface, to prevent the camera and crew's reflections.
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.
Visible crew/equipment: After Barney tells Andy he overheard Dr. Benson and Ellie "talking marriage" while spying on them, Andy heads over to the drugstore. Then, when Barney enters the jail cell, we can see a crew member's fingers (at the lower right edge of the bars, behind the wall) push the cell door closed. (00:19:45)
Revealing mistake: The woman in the foreground, as the siren is stuck, is carrying a paper grocery sack from "Hollywood Ranch Market" which was located on Vine St and Fountain Ave in Hollywood, Los Angeles. (00:09:35)
The Church Benefactors - S8-E20
Visible crew/equipment: At the start of the finance committee meeting, when Clara walks into the office modelling the choir robe, the shadow of the boom mic is visible on the door, in the top right corner of the screen. White tape marks are also visible on the floor, as Clara twirls in the choir robe. (00:08:15)
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.