Revealing mistake: At the party where Inspector Gadget lights a cigar for Scolex, he uses his thumb which flips open to reveal a flame. When they cut to the close-up of Scolex getting the cigar lit, you can see that there is a hand holding up a traditional lighter, and not Inspector Gadget's thumb any more.
Revealing mistake: When Robo-Gadget is rampaging through the city pretending to be Godzilla (a nod to another film Matthew Broderick was in) you see a shadow on a wall that is supposed to be Robo-Gadget's hands doing the Godzilla thing. You can tell from the shadow that whatever is making it is actually on rods.
Revealing mistake: After Inspector Gadget and Brenda jump from Dr. Claw's helicopter at the end of the movie, they land on the ground thanks to Gadget's trusty parasol. Meanwhile, Claw is up in the helicopter trying to keep it from spinning out of control. A few minutes later, we see him also land on the ground with a parachute. What exactly happened to the helicopter? We never get to see it crash or anything. It simply vanishes, never to be seen or mentioned again.
Continuity mistake: When Evil Gadget jumps out of the chief's office, he is seen falling forward, but in the next shot is seen falling backward just before he lands on the sidewalk.