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.
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.
Revealing mistake: When Andy, Barney, Gomer, and the kids are at the campsite in the woods, during the daytime scenes the only shadows that should be seen are the ones being cast by the sun, but there are shadows being cast in different directions due to the set lighting.
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).
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.
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.