In the first scene of the film, Fuller writes the letter with a fountain pen. This is clear in the close-up where the "feed" is visible under the nib. It also has a cap. The pen is dipped into a desk inkwell. Fountain pens are never dipped into an inkwell. It is set down in a grove on the desk set, which would be a correct action for a dip pen, which it was not.
In the climactic scene, when Craig Bierko prepares to shoot Gretchen Mol, the sound of the pistol cocking can be heard. However, having already fired several rounds, there is no need for the pistol to be cocked.
When Douglas is in the 1930 reality and in the pool of the hotel he grabs the revolver and he can be seen to check if it is cocked, which it already is. Then he places the barrel in the bartenders mouth and threatens to shoot him. It's believed that when the bartender pushes the barrel of the revolver out of his mouth and Douglas pushes it back in that he is re-cocking the already-cocked gun. However, if you watch after the bartender pushes the gun from his mouth, Douglas moves the hammer of the gun back into place (because he believes he's going to get his answer from the bartender). This makes re-cocking the gun necessary when the bartender doesn't answer after all.
In the opening sequence when Fuller enters the restaurant and is asked if he wants his favourite table, he spends a long time consulting his wrist watch (to see when his download time is up?) but he can't see his watch as it is hidden below the starched cuff of his shirt.
The goof item below may give away important plot points.
Incorrectly regarded as goofs
The front page of a newspaper shows the date as Monday, June 21, 2024; in the year 2024, June 21 will fall on a Friday, but since our world is just a simulation, and you can see this newspaper in the "real" world, it is possible that June 21, 2024 is a Monday.