The envelopes for the letters from Doug and from Pulver's friend that announces Doug's death have postage stamps. Postage stamps are not required, nor are they available, for serviceman in the combat zones.
When Ensign Pulver is talking to Lt. Gerard in the hospital, through the window to their left you see a blue automobile pass by in the distance. It is obviously a 50's style automobile although the film is set during WWII.
The shore patrol arrives with a truckload of guards to restrict ship's personnel from leaving "The Bucket" and Schlemmer drives a motorcycle off of the dock. Before he does so, he passes the ship and the truck and all the shore patrol personnel are missing.
The backgrounds behind both Roberts and the Captain vary when the Captain is confronting Roberts about the transfer request letter. The sky behind both Roberts and the Captain varies from clear blue, to having clouds, to being completely cloudy. There is land in the background behind the Captain's head that disappears.
The sailor on the LCM wants to trade the film, "The Sheriff's Daughter" with Hoot Gibson, to Mister Roberts. However, although there are five silent films entitled 'The Sheriff's Daughter" and two silent films entitled "The Daughter of the Sheriff", none of these feature Hoot Gibson. It is also unlikely that sailors would be watching silent films. The Hoot Gibson film nearest in name is "The Marshal's Daughter", but this was not made until 1952.
Just before Roberts leaves the ship Dolan tells him the Captain has had a new palm tree dug up and placed a 24-hour guard around it. But when Pulver later goes to throw the palm overboard, there is no guard anywhere near it.
When Mister Roberts finally leaves the U.S.S. Reluctant by seaplane, the plane makes a 180 degree turn to the left and makes a close fly-by along the starboard side of the ship. This shot was not taken from the plane; it was taken from a camera boat passing the ship. You can see the wake from this boat at the left side of the screen as it meets the side of the Reluctant at the waterline.
The goof item below may give away important plot points.
In the letter announcing Mr. Robert's death, it states that he and another officer were having coffee in the wardroom. If the ship were under attack, it would have been at general quarters. Mr. Roberts and the other officer would have been at their battle stations and no one would be lounging around. The destroyers on picket duty off of Okinawa and Iwo Jima tended to be at general quarters virtually from before sunrise to after sunset. While much is made of the carriers that took kami kaze hits, the fact is that the picket destroyers took a lot worse.