Plot hole: Live 50th anniversary episode, aired December 9th 2010 - When Sally is allowed into the rubble to comfort Molly as the firemen attempt to free her, one of them says they'll get Sally a hard hat. When we see her talking to Molly in the rubble, she has no hard hat on and instead one of the firemen is hilariously holding his hand above her head.
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: In the episode "Scheming Schemers" (1956 - fake Shemp episode) at the start of the episode, Moe is given the address for a plumbing job over the phone and he and Larry take off in their jeep - accidentally leaving Shemp behind. Next scene, Moe and Larry are in the house working on their plumbing job and minutes later Shemp just shows up outside in a separate vehicle. Shemp would not have known where the job is since he was left behind and never would have known the address Moe and Larry went to.