When Vivian is telling Marlowe the cover story for Carmen for the night before, her mouth does not match what she's saying. This mismatch is because the movie was re-edited after Humphrey Bogart and Lauren Bacall became a popular team; To Have and Have Not which brought them to stardom was released after The Big Sleep was filmed but before it was released. Scenes were re-shot, some new scenes were added and others dropped. The sequence was also changed to accommodate some of the new edits. The audio/visual mismatch is because the scene was kept but minor dialog had to be changed to make the re-sequencing track properly.
Marlowe keeps 2 guns in his car - under the dash. One's a snub-nose revolver and the other has about a six inch barrel. He uses and loses the snub-nose. Later, at Art's garage, he reaches in and grabs the larger gun. But in the exterior scenes following, he's got the snub-nose in his hand.
When Marlowe enters the Sternwood home, bringing Carmen Sternwood home from Geiger's house, he is not wearing his trench coat. But, as he leaves, he is given the coat by Norris, the butler, and puts it on.
When Marlowe returns to his apartment after taking Carmen home from Geiger's house, there is a tight shot of his (Marlowe's) front door, on which there is a bracket holding a card that reads "Philip Marlowe - Private Investigator" and "206" in large numbers. When he opens the door to the apartment and the shot changes to a wide one, the number has disappeared.
In the public library scene, a close-up shows Marlowe copying names out of book opened to new chapter with much white space on page, but in the long shot immediately following, he's copying from a full text page.
When Marlowe enters the window of the house by kicking it open, from the outside the window is completely covered with wooden slats, but from the inside it is a regular glass paneled window (that would have likely smashed if kicked open).
When the station wagon pulls out of the alley behind Giegers book store, the crates extend past the car body and are on the tail gate. When the station wagon pulls up to the Randall Arms and pulls into the alley, the crates are now fully inside the body of the car.
When Marlowe goes to Geiger's house, in the foreground is the house's mailbox with "AG Geiger 460" visible on it. When Marlowe later calls his pal Bernie at the D.A.'s office, he gives the address as "7244 Laverne Terrace".
When Bogart is in the house with the beaded curtain, the curtain is always completely still even after a very short time of passing through it. The first time Bogart goes into the bedroom and out again, you can see the curtain continue to move for a certain duration. His movements past the curtain should have them moving each time he comes back out of the bedroom because that duration is less than the first time.
In the public library, a close-up reveals Marlowe is copying information out of a book opened to new chapter, with a large margin at top and a chapter heading in bold face type. In the long shot a second later, the book is opened to random page of dense text.
When Marlowe is tied up and sitting on the floor, Vivian removes his white handkerchief from his black jacket. She then puts it back in the pocket, leaving about an inch of the handkerchief exposed. For the next minute or so, the jacket is shown but the handkerchief is missing. When Vivian unties Marlowe and in the scenes immediately following the handkerchief is in the pocket again with part of it exposed.
Just before the owner of the Acme Book Store draws the blind and locks the door, she is shown sitting on the edge of a table. When she is shown from the front, she holds a pencil in front of her and fidgets with it using both hands. When the camera angle changes, both hands are by her side grasping the edge of the table.
When Mr. White asked Marlowe, "your willing to lose your license for the case" Marlowe responds by looking directly at him, but when Mr. White asked him another question, the camera immediately cuts to Marlowe looking the other way (as if he never looked at Mr. White in the first time).