When the truck driver honks his horn, the sound of an air horn is heard. In order to sound that horn, he would have to pull a cord. Pressing on the steering sounds the city horn, which has a different sound.
When Professor Xavier is pushed to the air from his wheelchair by his child, his legs are also shaking.
After Stephen Hawking is thrown out of his chair into the bee hive, he is talking on his computer voice. He can't do that without the equipment built into his chair.
When Dragonfly learns to fly, he sprouts wings, which would presumably tear through his costume. His back has no rips or tears in later scenes.
When Dragonfly and Jill are kissing in the rain, their noses switch places from shot to shot.
When Rick Ricker checks his symptoms on the Internet, the dialog box shows the question "It's okay, you can tell me the truth. I'm a Doctor" with no options. The camera switches to Rick's face, then back to the computer with an option on the screen.
When searching for a $300 car, the search engine comes up with a "Nissan Shitbox". The car in the photo isn't actually a Nissan at all. It's (what's left of) a Peugeot 309.
Incorrectly regarded as goofs
When Hourglass loads software to destroy the world, he puts a Windows CD into his Apple computer. He may have been running a Windows emulator, which allows Macs to run Windows software. Recent versions of Mac OS include a Windows emulator.
When Dragonfly slams a thug's head on the brick wall, the wall gives slightly, revealing it as a softer, rubber-like material.
During the YouTube scene, the video time counts up by minutes, not seconds.
When Dragonfly jumps off the Gargoyle perch, it gives slightly, revealing it isn't real stone.