In the beginning, Fred is piloting a P-40 "Flying Tiger" Warhawk in pursuit of a low-winged enemy plane. After he shoots, a smoking plane is seen going down, but it is a biplane, not the one he was chasing.
When Fred's escort in the parade shows him to the door at the hotel, she turns as the hotel room doors open. In the next frame, they are back together in their original positions before the door opened. This is supported by the later photo of them that Phil Harriman shows Fred at the bar after creating the Flying Tiger drink.
Before entering Phil's office, Fred buys a golden delicious apple. When he enters the office, he has a red delicious apple.
Joan (played by Joan Leslie) refers to polyps as "little fish", when they are actually a structural form of cnidarians. Common polyps include Coral, Sea Anemones, Hydras, and Jellyfish.
After renting his apartment in Joan's building, Fred marks off the days on a calendar for the month of June. There is no year visible, but it fits June of 1943, the year this film was made. If Fred had just gotten out of the Flying Tigers, it should have been June of 1942.
Fred Astaire plays a WWII fighter pilot, although he was 44 years old at the time. A typical age for a WWII fighter pilot was half that. Most 44-year-old men who were serving in WWII did so from behind a desk.
During Joan's photo tour of the docks, the docks are gated and patrolled by security as unloading of a ship goes on, but Fred manages to get aboard ship and borrow a dolly without anyone seeing him.