Carol's hairstyle is often different in closeups, losing the curls that are seen in wider shots.
(at around 58 mins) 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 had been missing from their sockets now are present.
When Cliff takes Jeannie/Carol backstage to impress her with his drumming skills, he clearly is 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.
When Carol holds the drawing of the hat the hands shown holding the drawing have on dark nail polish, but Ella Raines is not wearing dark polish.
(at around 5 mins) At the theatre, a loud and isolated cymbal crash is heard, but the percussionist/kit player does not hit the cymbal, though his hand is moving near it after the sound. [The plot-reason for this isolated cymbal crash is that the drummer (Cliff) is distracted by the "Phantom Lady" in the audience.]
(at around 51 mins) Near the end of the prison visit, Carol 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.
Richman and Burgess investigate and requestion 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 requestion the taxi driver who saw Henderson with the lady.