The amount of oil and dirt on Courtney's face changes several times between scenes. The most obvious is when he goes up stairs to console another pilot who has lost a friend. As he goes up his face is slightly dirty, as he enters the room it is noticeably dirtier and when he comes back down it is much cleaner.
When Captain Courtney is rescued, he jumps on the wing and hangs onto the strut. When the actual aircraft takes off, not only was dummy used much further forward on the wing than Captain Courtney was, but it is an entirely different plane - a two seat trainer.
(at around 1 min) Phipps is writing a letter of condolence to the mother of a recruit who has been killed in action. As he stops writing to talk with Major Brand, he switches his pen from his right hand to his left hand and removes his glasses with his right hand. In the next shot, his pen is back in his right hand and his glasses are in his left hand.
In the beginning the planes shown landing and the planes that taxi up to the hangers are different.
The amount of engine oil on Flynn's face varies between shots.
Almost all the flying aircraft that don't crash are the Travel Air 4000, which was produced in USA from 1926 to 1929, so could not have been in use in WW1. It was produced for private flying and was never a military aircraft. Many of the aircraft seen on the ground are the Nieuport 28 - which was a WW1 fighter, but was French (and was not flown by the British) and did not fly until 1917 (the film is set in 1915). (Most/all scenes of actual flying and many aircraft scenes on the ground are lifted from the 1930 film.)
During the final dog fight when the aircraft is shown nose up, you can see a large, modern (for 1938) airport under them. It's visible in two shots.
When Courtney and Scotty are approaching the enemy airfield you can see Scotty's plane superimposed over Courtney. Obviously the film of the two is stacked and rephotographed to create the composite.
The opening shot identifies the year as 1915. From the first, the song played over and over on the phonograph is "Poor Butterfly." Sources seem to agree that this song was not published until 1916, and not recorded until later still.