When Jack, Carol, and Burgess are in the dressing room, there is only light from the few bulbs around the makeup mirrors. Right after Carol leaves, the lighting of the entire set changes and the room is much brighter with light coming from several different directions. Also, some of the bulbs around the mirrors that were previously missing from their sockets are suddenly present.
Carol's hairstyle is often different in close-ups, losing the curls that are seen in wider shots.
When Cliff takes Jeannie/Carol backstage to impress her with his drumming skills he is clearly sweating and his bow tie hangs loose. After he finishes and they both leave, his bow tie is perfectly fitted and he doesn't seem sweaty.
Early in the film, the character Scott Henderson is found guilty of second-degree murder and condemned to death. In New York in the 1940s, the jury would have to find him guilty of first-degree murder in order to condemn him to death.
During the theatre visit, near the beginning of the film (c.5 minutes), we hear a loud and isolated cymbal crash but the percussionist/kit player does not move at all at this point. It is inconceivable that a pit orchestra would employ a second percussion player and the plot-reason for this isolated cymbal crash is clearly that the drummer (Cliff) is distracted by the "Phantom Lady" in the audience.
Elisha Cook Jr's "drumming" at no time matches the drums on the soundtrack.
Near the end of the prison visit (after nearly 50 minutes of the film) Carol clearly mouths several words but her voice is not heard.
During an overhead shot in the sequence where Kansas trails the bartender through a deserted street after he gets off work, the actors are clearly connected waist-to-waist by a thin wire - most likely so that the actress stayed an exact number of feet behind the actor to ensure both were in focus during what was apparently a tricky camera set-up.
How the missing witness saw the hat that she had copied remains a mystery. Only one hat was made, the witness did not know whom it was made for, and the client wore it only in a stage show which the witness had not seen.
Richman and Burgess investigate and re-question the witnesses that Henderson claims saw him with the "phantom lady". While visiting Marlow in his apartment, Richmond complains that getting Monteiro to come clean is her last hope of proving Henderson's story. But she and Burgess never re-question the taxi driver who saw Henderson with the lady.