Molly enters the ladies' room with two bottles of peroxide and comes out a platinum blonde, with her hair rigidly set in rows of waves. Not unless a hair-waver and a dryer also happened to be in that ladies' room.
When Molly arrives at Ralph's house, the butler tells her he was going to deliver the note to her hotel in the morning. That makes no sense if she was expected at the house that evening.
In the opening scene in the park, the camera dolly tracks are clearly visible in the grass.
Scotty is supposed to be a hot-shot reporter, and yet: He knows her name is Molly, her photo is in the papers, but it never, ever occurs to him that she might be THE Molly.
The title character's name is misspelled "Molly Louvaine" in a newspaper headline.