The newspaper in the opening scene has a large headline saying "Dodgers Lose Again." But that wouldn't have been a big deal for a Los Angeles paper in the 1940s -- the Dodgers didn't move there from Brooklyn until 1958.
When Rigby is talking on Juliet's phone to her sister Leona, the handset and coiled cord to the phone are from the 1960's-80's, and not the straight cloth-covered phone cords in use in the 1940's when the story is taking place.
When Rigby (Steve Martin) is exchanging a $5 bill for a $1 bill with Ray Milland through the cracked open doorway, you can clearly see that the bill is marked "Series 1969" which is about 25 years after the time the film is set in.
In the scene with the pigeons after Juliet leaves, Rigby says "after a half bottle of bourbon..." and Jack Daniel's Tennessee Whiskey is pictured. Jack Daniel's is not bourbon - it never has been and never will be.
When Rigby is sitting in a chair with a drink while talking with Huberman (Ingrid Bergman)the film was reversed during editing resulting in Rigby appearing as a mirror image. Specifically, holding the glass in his right hand with his breast pocket and pocket square appearing on the right hand side of his suit, instead of the correct, left side. Between cuts, the image reverts to normal.