(Widescreen version only) The shadow of a boom mic is visible in the upper left portion of the screen as C. C. Baxter is rushing to open his apartment door after being alerted to an odor of gas by his landlady.
Fran buys a record album recorded by the small combo that performs in the Chinese restaurant she patronizes but when the record is played in Baxter's apartment, the tunes are lush orchestral arrangements, nothing like the lounge music heard in the café.
After Baxter gets home for the first time, he cleans up by placing a large waste bucket of empty liquor bottles outside his apartment door. An hour later when he has to vacate the apartment for one of the other men, the waste bucket full of bottles is gone. There was no-one around to carry his bottles away at that time of night.
After Bud finds out that Ms. Kubelik was with Mr. Sheldrake, he leaves wearing his new hat, leaving his old hat behind. Later that night when he arrives home with the woman from the bar, his old hat is on the hat rack on the wall.
When C.C. Baxter gets into Miss Kubelik's elevator to go to the 27th floor, Miss Kubelik pushes a button, presumably 27, which lights up. When they reach their floor, and the elevator doors open, rather than switching off, as an elevator button would normally do when elevator reaches it's destination, the button stays lit, and remains lit through the remainder of the scene.
The layout of Baxter's apartment makes no sense, especially in the context of Dr. Dreyfus's apartment. Dreyfus lives next to Baxter, which means their walls should be adjoining the full length of both flats. But from inside Baxter's living room one can see windows in both his kitchen and bedroom facing directly where the Dreyfus apartment should be (and there would likely be a window in the bathroom between the kitchen and bedroom). Dreyfus's apartment would have to veer immediately off to the extreme right when you enter it and be no more than a couple of inches wide in order to allow the kind of set-up seen in Baxter's apartment - clearly unrealistic, if not downright impossible.
Fran's brother-in-law comes looking for her at her employer's office building after she hasn't been home for 2 days, beginning Christmas Eve. Christmas 1959 was a Friday, so 2 days after she went missing would have been Saturday or Sunday, when nobody would have been in the office.
The goof item below may give away important plot points.
At the very end of the film, Baxter and Fran sit down to play a game of Gin. This game is played with 10 cards dealt to each player, but Baxter deals at least 13 cards to both Fran and himself before the picture finally fades out.