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.
The Ricardos Are Interviewed - S5-E7
Continuity mistake: When Ricky's new agent insults the apartment building Ricky explains that Fred and Ethel are their best friends, and we see Lucy drop the toy train onto the chair and take hold of the jack-in-the-box toy (in the crook of her left arm) with her right hand, but when it cuts to the closeup the toy train is back in Lucy's right hand and the jack-in-the-box toy is once again held in her left arm. (00:03:50)
Continuity mistake: When Ricky comes over to Lucy, Fred, and Ethel after he sings for the first time, his hands are by his sides. The shot changes his hand are now in his pockets.
Continuity mistake: When Lucy is talking about finding an undiscovered artist she is holding a French dictionary. The way she is holding the dictionary changes a couple of times.
Lucy Gets a Paris Gown - S5-E20
Continuity mistake: When Lucy and Ethel are sitting at the table at the cafe just before Ricky and Fred show up, Lucy starts to put her arms on the table. The shot changes and her arms are already on the table.
Return Home From Europe - S5-E26
Continuity mistake: As Lucy gets up from her airplane seat holding the big hunk of cheese wrapped in a blanket she is masking as a baby, it bends in half as if it were already cut. This is the scene before she sneaks off to the kitchen to cut it apart. In prior scenes, the cheese is intact and solid. (00:17:10)