Factual error: When Peevey gets stuck in the truck, Cliff fires up the rocket pack and simply pushes it clear. If the rocket pack can push out enough thrust to move a 2 1/2 ton truck, Cliff's body can't take that kind of stress. He stretches his arms out, starts pushing, and sets off the rocket, putting himself between the truck and an accelerating rocket demonstrably capable of moving it. The truck is stationary so has 2 1/2 tons of inertia. The rocket instantly accelerates it to an appreciable speed. Human arms cannot take that kind of stress. He's going to be crushed.
Continuity mistake: When Cliff is being chased by the mobsters in the fancy restaurant, he escapes by flying through a stained glass dome directly above the dance floor. Two mobsters below aim up at him and start firing as he crashes through the dome, but no glass showers down on them.
Plot hole: The helmet design is rubbish. The eyeholes are far too small - that close to the eyes the divider between the eyeholes would block almost a third of the wearer's vision. He would hardly be able to see what was right in front of him - and it is not too close nor too narrow to fall into the "blind spot" between the eyes. This is not a character error by Peevey - the helmet design came from the original plans - it is even shown in the German propaganda film!
Continuity mistake: When Cliff and Peevey first test the rocket pack by strapping it to a statue, the statue skids across the ground and crashes into a small mound of dirt on the otherwise completely flat field. But when they run over to collect the pack, the mound is gone and the statue is lying in a crater, as if it had hit the ground from a much steeper angle.
Deliberate mistake: When Cliff crashes into the pond after his first flight, Peevy tells him that he has to look around with his eyes and steer by moving his head. However, just before this as he is flying next to the passenger airplane, he looks over at it, turning his head to the left without changing directions at all.
Factual error: The only ways to steer a rocket are to gimbal the nozzle or put steerable vanes in the exhaust. You cannot, ever, put a rudder on the front of a speeding aircraft.
Continuity mistake: When he first flies the GeeBee, the joystick changes colors in the different scenes.
Deliberate mistake: With the rockets positioned on the rocketeer's back, their thrust would create an unopposed moment about his center of gravity. This would result in the flyer being slammed face first into the ground, not flying straight up. (Note that functional rocket packs, such as were flown briefly in "Thunderball", place rocket nozzles on either side of the flyer's torso, not both on his back.)
Continuity mistake: When Cliff first puts the Rocket on to save Malcolm, he doesn't do his helmet strap up. When they walk outside, just before Cliff takes off he turns back round to Peabody and his strap is done up.
Continuity mistake: When Neville Sinclair walks Jenny to the dance floor, the bandstand is empty. When we next see it, about 25 seconds later, all the band members are back. (01:01:53)
Revealing mistake: In the first shot of the club scene, where we see the clarinet up close, the ligature (the metal band that holds the reed in place) has been placed upside down. This would make it extremely difficult, if not, impossible for the clarinet to play a single note. So, the "clarinet player" physically could not play his instrument, even if he wanted to. (00:51:00)
Other mistake: When the FBI agent says "Maybe it's time to get a real job" his partner can be seen mouthing the same words.