When Dave finds out about the Italian team coming to race in Indianapolis he reattaches the front wheel to his cycle, leaving the quick-release lever pointing forward. However, in the next shot with the postman, the lever is facing to the rear.
When Dave is drafting behind the Cinzano semi-truck, his bike is on the small chain-ring and he is managing to travel at 50+ miles per hour. An earlier shot shows him in the large, and correct, chain-ring behind the semi.
At the end of the movie, Dave and the French girl are riding their bicycles away from the music school towards the main library. Then when they ride past Dave's dad, they are in front of the music school, about 400 feet back from where they were in the previous shot.
Immediately after Mike dives into the quarry and swims into the refrigerator the scene switches to Dave and Moocher talking. The quarry surface is clearly visible in the background, but shows no ripples despite Mike's dive.
In the Little 500 race the bikes are single-speed with a coaster brake and small gear wheel on the rear hub. In close-ups of Dave pedaling, however, the upper and lower halves of his chain are parallel, indicating the presence of a dérailleur just beyond the edge of the frame.
When Dave first collides with another bike during the race and crashes, he has blood and dirt on his leg. When he eventually comes into the "pit area" to make an exchange, there is no dirt, blood or marks visible on his leg.
The scene were Dave is chasing the Cinzano truck towards Indianapolis, he passes fields of corn at least 5-6' tall. As this occurred prior to the Little 500, which is held in early spring, it is doubtful the corn would have grown to that nearly harvest stage in early spring.