Plot hole: How does Lucy get back up to the apartment bound and gagged? It would seem impossible to hop up all those steps, possibly take several hours if not all night.
The Million Dollar Idea - S3-E12
Character mistake: When Ricky sits down to calculate by hand how much Lucy and Ethel have spent on supplies to make and sell their salad dressing, he adds $7.21 for the groceries, $1.20 for the jars, and 10 cents for the labels. He says his total is $8.31. He miscalculated; the correct total is $8.51. Ironically, his profit calculation is correct at 3 cents a jar. (00:15:10)
The Benefit - S1-E13
Revealing mistake: There's suddenly an upright piano in the living room where Ricky's desk used to be.
Suggested correction: There's nothing sudden about it. From the start of this episode, the card table is at the center of the living room, with the couch pushed back toward the wall, and the desk is behind the couch (holding telephone, etc), up against the wall where the piano usually is in other episodes. Ethel asks Ricky about the sheet music on the piano, and he explains he's been working on a musical number for the show at the Tropicana. So, for this episode's ongoing musical storyline, the positions of the piano and desk have been switched. This occurs again a few episodes later in Breaking the Lease. The rearranging of furniture between episodes occurs often, and would only be a problem if it happens within a scene.
The Adagio - S1-E12
Character mistake: Lucy turns in for the night in full makeup and lipstick, instead of washing up.
Character mistake: When Fred, Ethel, and Ricky confront Lucy on what she wrote about them in her novel Ricky states "I'm so hammy I should go lie down between two slices of rye bread." This is incorrect. Lucy wrote "He turned into such a big ham you could stuff him with cloves."
Continuity mistake: The Mertz' check out of the hotel and are in the garage on a motorcycle. Later they are back in their hotel room with Fred napping in a chair with a hurt shoulder.
Suggested correction: Ethel and Fred checked out of room 317, intending to leave. Later, after their motorcycle crashed, Ethel and Fred checked back in and were given room 372 (time codes 00:09:05 and 00:12:15), so Fred (with his hurt shoulder) is napping in the chair in his and Ethel's new room. There's no continuity mistake.