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.
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.
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.
The goof items below may give away important plot points.
Incorrectly regarded as goofs
When the bartender goes home after the saloon closes for the night, the New York streets are deserted and the subway platform is virtually empty, exactly as you'd expect for such an early hour of the morning. Yet despite when he is struck by a car upon arriving home soon afterward, the sidewalk outside his apartment is filled with at least a dozen people standing around chatting, sitting on the stoops, etc. - none of them in pajamas or nightclothes. (In fact, in the days before air conditioning, on a hot summer's night, the stoops of apartment buildings would be crowded with residents who couldn't sleep because of the heat and grabbed a breath of cool morning air while being social. Only New Yorkers over the age of 60 would know this.)
The crux of this murder mystery involves a hot-headed Broadway diva who becomes so enraged when a woman attends her revue wearing an exact duplicate of the bizarre highly-distinctive hat she sports during an onstage musical number, that she immediately destroys the hat after the performance and later falsely tells police she would never wear anything but a one-of-a-kind creation, even though this misinformation (the theater-goer wearing the copy of the chapeau provided to be an alibi) will erroneously send a man to his death for a crime he didn't commit. In reality, police could have cleared up this point immediately had they simply questioned the show's costume designer or even checked the credits in the show's program to find the milliner who created it - performers never wear their personal wardrobes in elaborate musicals. Furthermore, the show in question was a long-running hit, suggesting there would have been publicity pictures of the star wearing the hat - or, if anyone had bothered to alert newspapers, thousands of theater-goers who could vouch they had indeed seen her wear it onstage during earlier performances.