In the scene where Zsa Zsa Gabor is laughing hysterically at how much money she is going to take her husband for, the abrupt cut shows her sullen for about ten seconds before she starts to laugh again.
On the day of the Mrs. Mississippi pageant, and the day Jeff receives the governor's letter, Annabel's manager says the Mrs. America and Miss America pageants are happening in Atlantic City in one week. However, the governor's letter was dated June 11, 1951 and presumably received within a few days. The Miss America and Mrs. America finals in 1951 occurred in September, much more than one week away from June 15th or so.
Melvin Bush's registry book showed Hector's address as being in Lanhasset, New York. The address on the envelope from the governor shows his residence as being in Manhasset, N.Y. If this goof was made by the set decorator, then it's a factual mistake, but there's a slight chance it was an intentional character error given how scatter-brained Mr. Bush was.
When the Gladwyns are shown in the back seat of their car being driven to the studio, it's supposed to be raining heavily outside, but the cars seen in the rear projection are not using their windshield wipers.
The letter inducting Willie Fisher into the Army carries the date "May 11, XXXX", not showing any year, as is of course normal. The letter also refers to the "Asiatic-Pacific theater", when "Asia-Pacific" would be the correct term.