Michael describes Patricia as having golden hair when she's very obviously a brunette.
After Michael and Patricia are shot at during a foggy night, the film resorts to the expository trope of a newspaper headlines montage. Five different headlines summarize the evolution of the police investigation over what would seem like a period of days (judging by the changing articles elsewhere on the front pages). However, after the montage, the story resumes the next morning, about only 10 hours later.
The calendar in Casey's shop is royally screwed up. The date the inspector asked Michael about was correct for 1937. The calendar on the wall shows January beginning on a Friday, which is also correct for 1937. However, the following February begins on a Tuesday when it should have begun on a Monday. Most of the other visible months begin on the wrong days for 1937, or any year. For example, April is shown to end on a Monday, followed by a May that begins on a Friday.
How did the Umbrella Man find the envelope he wanted in the postman's bag so fast? There were only 8 seconds between the postman groaning from the Umbrella Man's attack to when the inspector ran up, finding only the postman. It would take at least a few seconds to grab the bag, open it and dump the contents on the ground, then at least a few more seconds to scan the dozens of envelopes to find the one with the correct address. That would leave the thief no time to get out of sight.
When Michael first meets Arthur, he says two men have been killed, but at that point in the story he only knew from hearsay that a man was missing and a policeman was dying from a gunshot. No deaths had been confirmed. A reporter would know better than to jump to conclusions.