Revealing mistake: In the final scene when the escape pods land in the water, it's visible that the splash consists of large drops, which would never appear in normal conditions but is easier to do as a digital effect.
Continuity mistake: When Sky Captain and Polly go visit Dr. Jennings, it is pouring down rain, but when they enter his basement laboratory both of them are completely dry. Granted, they were only exposed to the rain for a few seconds in the distance from the car to the building, but they would still have gotten least a little wet.
Deliberate mistake: In several scenes, Sky Captain chases the enemy through the streets of New York. At even the slowest speeds to maintain flight, the hairpin turns around street corners would generate forces equivalent to 15 to 20 times the force of gravity. Even if the aircraft could survive these forces, the pilot and passenger could not.
Factual error: When Polly Perkins goes to Sky Captain's headquarters, she promises not to take any pictures without his permission, then proceeds to take a picture, holding her camera behind her. Before you can take a picture with an Argus 35mm camera, which is what she has, you first need to flip down a lever on the front of the camera that "cocks" the shutter. When you trip the shutter, the lever flips back up. But when Polly takes the picture, you hear the shutter "click", but the lever doesn't flip up.