As Buster goes to the newsreel office for the first time, the same girl leaves the building in consecutive shots.
At the start of the "date", Buster is wearing white socks and does so through most of the date. But in the famous changing-room scene, and in another place or two during the date, he's wearing dark socks.
When Buster breaks the glass in the door for the first time, the remaining pieces change in between shots.
When Sally's boss learns she gave the "tong war" tip to Buster instead of her boss' crew, he speaks to her harshly. She puts on her hat, as if she's been fired and is leaving. Immediately afterward though, she is sitting at her desk, sans hat.
During the Yankee Stadium sequence, an elevated train goes by in the background in one shot, but disappears before leaving the frame in the next shot.
The "ticker-tape parade" shown at the end of the film shows little, if any, ticker tape. (Ticker tape has a distinct spiral shape.) Instead, the ticker tape shown is nearly all small pieces of cut-up paper.
At the end, when Buster and Sally are walking in the ticker-tape parade that Buster mistakenly thinks is for him, it can be seen that the parade is actually for Charles A. Lindbergh after his historic flight over the Atlantic which took place in 1927. Earlier, after Buster had purchased his movie camera, his bank passbook noted that the account is closed on June 30, 1928.
However, this is not a Goof, as the parade is used simply for the effect of the movie, not as a historical representations.
Right before the last time the glass in the door is broken (when the wind shuts it abruptly), it can be seen that it's already cracked to ensure that it will break when the door slams.
Near the end, when Sally and her Boss are watching the part of the tape filmed by the monkey, the shots of Luke bringing Sally to land change, as if the monkey had moved the camera. There's also some movement to fit the characters better, which couldn't be managed by an animal.
Buster's movie camera is missing a lens throughout the picture. One scene during the Tong war sequence, the shutter can be seen spinning inside the camera as Buster is turning the crank.
It's unlikely a New York City office would have a Union Pacific RR calendar on the wall. The Union Pacific, which was headquartered in Omaha, did not operate east of Chicago at the time.
While Buster is filming the boat race, Sally and Stagg get thrown from their boat, and Buster leaves his camera to take a boat out to rescue Sally. As he leaves the beach, it is seen that the monkey is not cranking the camera. While the monkey might have cranked the camera for a while, it wouldn't have continued doing so after Buster got back to shore with Sally.