Factual error: Andy tells the hobo "that will be the 3:45 stopping for water", meaning its a steam locomotive, but by the time this episode was made railroads had long since switched to diesel.
Continuity mistake: At the beginning of the show, Andy notices a wanted poster on the office bulletin board and tells Barney that he had learned from news accounts that the wanted person had been captured. Barney removes the poster and places it in a file cabinet. The next day when Roger reports for duty, he and Barney are looking at the bulletin board and the wanted poster is back up.
A Black Day for Mayberry - S4-E7
Factual error: An armored truck is going to Fort Knox from Denver would never go through Mayberry. Fort Knox is in Western KY south of Louisville on I-65. NC is accessible mainly by going south down I-75 to Knoxville, then heading South East along I-40 towards Asheville, NC. To get to NC from Fort Knox you would need to go up to Lexington then Knoxville/or go east across the wky fka The Green River pkwy to I-75 then to Asheville. There is no easy route; why would they go through KY to NC and turn back to KY?
My Fair Ernest T. Bass - S4-E17
Character mistake: During the party at Mrs. Wiley's, Ernest Bass' girlfriend is called "Ramona." At the end of the show, Ernest T keeps calling her "Romina".
Goober Takes a Car Apart - S5-E17
Factual error: How in the world did Goober lift that heavy motor out of the car?
Barney and the Cave Rescue - S4-E13
Plot hole: Andy and Helen go into the cave and there is a cave in. They find their way out and go into town to change, when they hear on the radio that Barney is getting the townsfolk together to form a rescue, Helen and Andy go back to the cave so barney can rescue them. This makes absolutely no sense. Once they get out of the cave why didn't they go back to tell Barney they were out, why did they have to flag down a truck for the ride back into town? Did they get to the cave with the squad car? It's hard to believe nobody saw them back in town, and how did they get to the cave?
Factual error: A town like Mayberry with a mayor and council would have its own police force, not just a sheriff.
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.