Revealing mistake: When Lucy is cutting up the fake fur coat to scare Ricky, it is clear that the parts that she has to remove are already cut and just loosely sewn in place.
Lucy Is Jealous of Girl Singer - S1-E11
Revealing mistake: During the rehearsal for the Jezebel number, the dancer's skirt trim is already partially torn or unattached to make it easier to tear, and it is clear that Ricky purposely steps on the trim to rip it off the skirt.
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. The rearranging of furniture between episodes occurs often, and would only be a problem if it happens within a scene. From the very start of this episode, the card table is already 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.
Revealing mistake: When Fred and Ethel come with parts of their apartment's ceiling on them look closely at Ethel. The piece on her left shoulder falls off revealing what looks like a velcro pad to keep it in place.
Revealing mistake: When Lucy's pinned against the sink by her bread Ethel goes for a saw. If you watch the doorway you can see her shadow where she's being given the saw.
Revealing mistake: When Ricky brings home the psychiatrist, in all the long shots of the living room from the time Ethel walks around the coffee table toward Fred, we can see that all four coffee table legs have been propped up on stone blocks.