Visible crew/equipment: When Flip pays a visit to the Taylors, he and Andy walk over to Opie, and the actors' two tape marks are visible on the floor between Opie and Flip. (00:21:05)
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)
Character mistake: Barney complains about Andy escorting the Apricot Queen around Mayberry many years prior. Barney had hoped to be her escort. He remembers the date, telling Andy it was "June 23, 1952, a Saturday afternoon." June 23, 1952, was a Monday.
My Fair Ernest T. Bass - S4-E17
Other mistake: Barney says, "I brought the car to drive you over to Mrs. Wiley's," but later when Andy, Barney, Ernest T., and Ramona leave Mrs. Wiley's we see that they're at the house that's actually next door to Andy - though to hide this fact there's a station wagon in the driveway, and they've removed the trellis beside the door.
Other mistake: The safety goggles Barney wears is missing both of its lenses. While it's possible that Barney may wear the goggles just for the look, that possibility is negated when Barney states, "Every piece of equipment and wardrobe has a specific function: helmet - anti-crash; goggles - anti-foreign objects in eyes, that is gravel and/or bugs."
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.
Other mistake: When Barney, Thelma Lou, Helen, and Andy are in the squad car talking about getting dinner, there's black tape on the chrome trim of the triangular vent window - presumably to prevent the camera/crew's reflections; and the vent window is shut in wide shots, but open in the closeup - also to prevent reflections.
Continuity mistake: In the exterior shots of the bank's doors there are rectangular door knob backplates on the outside of the door, but in the interior shots when the door's open we can see the door knob's backplate is narrower and has a decorative top.
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.
Other mistake: In this episode (aired Sep. '64) Barney tells Andy that 5 years ago on May 16 (1959) he joined the sheriff's office, and later they all have a surprise party for Barney. However, in season 4 "Citizen's Arrest" (aired Dec. '63) Andy tells Barney that he found, "The custody receipt for the first revolver I issued you... August, 1953... 10 years ago last month since I swore you in." Andy even mentions having a party with a cake that has 10 candles on it.
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)
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.
Andy and the Gentleman Crook - S1-E21
Character mistake: When Barney and Gentleman Dan play gin, Barney deals only 8 cards. In gin, the proper amount would be 10.
Revealing mistake: Before Barney jumps out from behind the crate to surprise the two cops in the alley you can hear the director yell out "Don".
The Barbershop Quartet - S7-E3
Other mistake: At the end of the show Andy talks about the upcoming bowling league. Howard shows Andy his injured left thumb. This should not affect his bowling since Howard is right handed.
Other mistake: Opie accidentally drops and breaks the bottle of perfume and doesn't realise it's colored water, a bottle of perfume would smell for days.
Other mistake: In some episodes Barney can sing beautifully, but in others he can't sing at all.
Andy and Opie, Housekeepers - S1-E23
Continuity mistake: Andy is sitting on the couch, next to him on the end table is a glass and what appears to be a dark bottle like a beer bottle. They cut away, when they return to Andy on the couch the glass and the dark bottle have disappeared.
Aunt Bee Learns to Drive - S6-E21
Other mistake: Aunt Bee says that Andy had to drive her to Clara's and Andy says it's not a bother, but Clara has always lived right next door.
Opie's Group - S8-E9
Visible crew/equipment: While Opie and his band are playing at the party there are shots of the kids dancing, and in the shot of a girl with blonde hair wearing a blue dress, the camera angle is low and we can see the top of the set. (00:10:25)
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.