Plot hole: When Barney's telling Andy about the poem written on the bank wall, Barney blames Opie and doesn't mention anything about Opie using chalk, but when Opie walks in Andy tells him, "Barney says there's a poem written on the wall of the bank, and that you were standin' along beside it with a piece of chalk in your hand," and then in a few moments Opie explains that some big kids pushed the chalk into his hands. So it's impossible for Andy to have known that chalk was used to write the poem, and not a marker, a pen, or a pencil.
Plot hole: About 2/3 through the show Barney is locked inside the armored car and Gomer puts about a gallon of gas into the gun port onto Barney's shoes. After Gomer is told of his error, Barney stays inside the truck. The fumes would have killed him, not to mention the risk of explosion.
Plot hole: When Barney and Andy are on the street watching the window painter, Andy looks over and says "I believe that's Briscoe Darlin's truck!" - but when they were driving up to his place the next morning, they both got out of the car and were wondering where their house was they heard music playing. The next scene shows them pulling in - seems that if he was able to recognize the truck in town, he would have been able to recognize the truck and they could have drove around looking until they found it?
Plot hole: In the episode where Barney confronts the two farmers for selling produce on the side of the road, Barney mentions that they're not allowed to sell because of "section 17.4" of the ordinance code - that street vending is not allowed, and Andy confirms it. but in the episode The Merchant of Mayberry (I think), Bert comes to town and is selling dry goods, etc. and he's allowed to sell. Even Ben states that there must be a structure if he's selling - wouldn't you think that Andy and Barney would have known that already before setting him up to sell?
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?