The hole in the roof of Prof. Quail's car disappears and reappears during the chase.
When Professor Quail Finishes his beer and takes off, his Autogyro is light-colored with a dark stripe along the side. A second later the Autogyro is shown in flight and it suddenly has a completely dark paint job.
W.C. Fields reaches from bed to turn on the lamp on the nightstand. The lamp goes on before his hand reaches it.
As Professor Quail is departing from Juárez (México), a large hangar behind his autogyro bears the sign, 'Aereo-Puerto-Juarez'. While 'Aereo Puerto' is a translation for the separate words 'Air' and 'Port', an airport in México would be called an 'Aeropuerto' (all one word with 'air' translated as 'aero-').
When the sailor jumps down to the stage at the beginning of the musical nightclub number, you can clearly see the cable that is attached to him to lower him to the stage. A second later, it is gone.
During the scene where Prof. Henry R. Quail is by his auto gyro talking to Doctor Wong and Peggy Hopkins Joyce, you can see the shadow of the boom mic moving above their heads. The boom mic then hits something, presumably the auto gyro, making a noise which makes Prof. Henry R. Quail and Peggy Joyce look up.