During the chase towards Sunset Studio Billie is driving her cab with Lawrence sitting in the back. When they get out at the studio gates Lawrence gets out from behind the wheel and Billie from the back. Presumably there was a scene where they switched places that ended up on the cutting room floor.
When the murderer holds a gun on the five main characters and dashes out of his office, he closes the door and then fires three shots through the door from the outside. One of the bullets smashes a bust of Shakespeare inside the office, but we see NO holes appear in the door while we hear the three shots. And yet, when the camera cuts away from the smashed bust and back to the closed door, suddenly it does have three bullet holes surrounded by splintered wood.
When the murder weapon is discovered, both the Falcon and McBride hold it by the handle with their bare hands, which would render a fingerprint analysis almost useless. Both should certainly know better.