When Carol meets Mr Miller for the first time after the big party, he's hung over and playing a bass. It is obvious that he's not actually playing it. Although we hear different notes being played, his left hand never moves and his right hand doesn't always match the rhythm.
When Carol takes Jerry to the beach, at the end of the scene Jerry has the blue towel in his hand as he lies back into his seat. As Carol lies back next to him into her seat, the blue towel is now draped over the back of both their seats.
The entire movie takes place during the one week the VIP's (Greek chorus) are in town, yet during that week, along with all the other things that occur, Jerry Webster (Rock Hudson) and Peter Ramsay (Tony Randall) go on a Canadian vacation and grow beards that would take at least three weeks to grow.
When "Rebel" (Edie Adams) testifies before the Ad Council, she says "Jerry" (Rock Hudson) considers the top of the Chrysler Building "Inspiration Point", because "it looks down on Madison Ave." The Chrysler Building looks down on Lexington Ave and because of buildings in the way, Madison Ave, two blocks away, is not clearly visible.
Jerry and Rebel both wrongly pronounce the Latin phrase, and the English subtitles misspell one word. The phrase should be Veritas et probitas super omnia. Both characters clearly drop the "p" and say instead, "robitas" which isn't a word.
At one point in the film, Jerry and Carol talk with an aquarium behind them, which holds a fish that is eaten by another as the scene ends. The "aquarium" is clearly just a short loop of film being played forwards and backwards in succession - the fish repeats the same two motions over and over until it is eaten.
The goof item below may give away important plot points.
According to the real Dr. Tyler, each "mint" is the equivalent of three martinis. If Carol and Jerry only had 3 mints each, that would be the equivalent of 9 martinis, resulting in a blood alcohol level of over .45 for Carol and over .35 for Jerry. This would render them both comatose and thus unable to obtain a marriage license, and no Justice of the Peace would marry a couple that inebriated. There would certainly be no way Jerry could impregnate Carol. All-in-all the ending of the movie just doesn't make any sense, even for a comedy.