Set in 1938 but mentions Deutschemarks, which weren't introduced until 1948. The German currency at the time was the Reichmark. The term "Deutschemark" was used for double-D alliteration because Campbell's line was "He pay you in dollars or Deutschemarks?"
In the first air show scene (and later in the nightclub), press photographers have 4"x5" Speed Graphic press cameras of a post-war model with the polished aluminum (shiny) lens boards. In 1938 they would have had black painted wooden lens boards.
The final scene shows a stop sign that is white letters on a red background. The White-On-Red color scheme was first mandated in the 1954 revision of the MUTCD, Manual of Uniform Traffic Control Devices, section 29.
When the Nazi agent is yelling at Sinclair, he shouts, "Ich habe meine Bestellung, und du auch!" ("I have my order, and so do you!") The German word for a military order is "Befehl." "Bestellung" is the order you give to a waiter.
After the GeeBee is hit by the fleeing thieves, the long shot shows oil splattered on the windscreen. Then a close up shot shows the windscreen clean before it is sprayed with oil, then it switches back to the long shot with oil on the windscreen again.
When Cliff nearly crashes into the truck when he's "landing", we also see him standing on the flatbed in the back of the truck (this is because it is the same shot as when they are making their getaway when Cliff is pushing the truck with the rocket pack).
After Cliff's Gee Bee R-1 racing plane is damaged, as he is attempting to land we see the engine temperature gauge indicating the engine is overheating. While the engine was shown being damaged, the R-1 was powered by a Pratt & Whitney R-985 Wasp Jr air-cooled radial and could not be overheating as airflow at the 200 mph it was flying would keep the engine cool. The oil pressure could have dropped, but the engine would not have overheated in flight - overheating would only be caused by running the engine on the ground for an extended period of time.
The word "commando" was not widely used until 1940 after the need for quickly deployed raiding forces arose after the fall of France. However, the word came from the Boer War, where it described Afrikaaner long range penetration units fighting the British, so its use in 1938 is conceivable.
When Sinclair ignites the rocket pack to escape the zeppelin, the rocket exhausts are aimed vertically yet Sinclair travels horizontally. Actually, he is slightly tilted, which does give him a slight horizontal push (given the pack's demonstrated power even the slightest tilt would do the trick). Moreover, the zeppelin itself is moving too, so Sinclair's horizontal movement is at least partly due to the relative motion to the zeppelin.
The goof items below may give away important plot points.
When Sinclair destroys the "LAND" of the "HOLLYWOODLAND" sign, steel frames are holding the letters. In 1938, the frames were wood, the steel was added only in 1978. Furthermore, the LAND letters stood until 1949, but that knowledge is part of the gag.
After The Rocketeer accidentally breaks the cable on the Zeppelin, the German pilot is unable to turn the steering wheel to the left. After Lothar crashes through the window, the steering wheel spins freely in that same direction. However, a few seconds before that, the wheel is spinning to the right before reversing direction. The turn to the right is what gave it the freedom to spin back, until it reached the stuck point again (which is clearly seen), so there is no mistake here.
About an hour into the film when the thugs attack the cafe asking for Cliff Secord, Cliff does not tell them who he is. Then, Cliff and Peevy run away and Peevy helps Cliff escape. Then Peevy is held at gun point, and we don't know what happens until twenty minutes later that we see Peevy, safe and sound, with Howard Hughes, as if nothing happened. The explanation is straightforward, however. The one who pointed the gun at Peevy was an FBI agent, who took Peevy to Hughes. The apparent plot hole is intentional, to keep the audience in greater suspense.