The timer on the bomb counts down to 1:02:52, in the next scene it is 1:03:00.
When Dr. Tam draws the design of the atomic bomb on the board, he only wipes away a small section of top-middle of the board (a circled ST can still be seen as he puts down the duster and picks up the chalk). When he finishes drawing the bomb design, everything that was originally on the middle of the board has been wiped and replaced with the bomb design.
While Lamont and Moe are talking about Margo Lane, the cab they are in is driving towards a building. Just as they reach the point where they must turn either left or right, we cut to a view out of the rear window. We never see the cab turning, so we have to assume it drove straight through the building.
Right after the sarcophagus opens, Khan removes his silver mask with his right hand, and in the very next cut the mask is gone from his hand never to be seen again.
During the investigation of the murder at the museum, a policeman working for The Shadow alternately wears his "Shadow" ring on his right/left hand as he leaves to send a message to The Shadow (the establishing shot shows it to be worn on his right hand).
When the waiter brings Miss Lane the '1928 Rothschild' not only is it already uncorked, but the label is wrong. No-one with any education would ever accept an already-opened bottle. A genuine Rothschild carried a different label entirely. The film-makers may have been aware of the deficiency as the label is never shown facing the camera.
The timer in the bomb used a display tube called the Nixie tube to show the digits of the countdown. The original Nixie tube was patented in 1938, but it looked like a small TV screen on which the numbers were displayed. The tube design used in the bomb was not introduced until 1962.
Khan should be cut and slashed by the flying shards of glass as they strike him, but he does not have a single cut anywhere.
While Moe's cab's design is based on a Cord automobile, it appears to have been built on a rear-wheel-drive chassis, based on its handling. The Cord was front-wheel-drive.
The Shadow doesn't turn invisible. He has the ability to "cloud men's minds." Meaning he may appear invisible to someone he uses his powers on, but he doesn't actually turn invisible.
Shiwan Khan says that he is the last living descendant of Genghis Khan, which is provably not true (see trivia). He is probably counting only the line of approved dynastic marriages, as opposed to by-blows from one-night stands and other "mongrel" branches.
The Shadow tells Dr Roy Tam, "Mr Schrevnitz here will instruct you in the way I will contact you if I need your help." The Shadow then gives the instructions himself, but he may have wanted Moe Schrevnitz to clarify his instructions.
When Farley Claymore goes mad in the hotel attempting to kill the Shadow, the torch in his hand does not move in time with the light on the ceiling.
Margo is at the Cobalt Club inquiring to Uncle Wainwright about what the police are doing about her father. Right after she says, "He spoke to me in Chinese" the shot over Margo's shoulder shows Uncle Wainwright choking on the last of his drink and you hear him put the glass down on the table. The next shot, from Uncle Wainwright's right side, shows he still has some drink left and again you hear him put it down on the table.
The two door guards with an elevator in front of them are shot with arrows. When the second guard gets shot, it's obvious that the arrow isn't fired but appears from under his coat (released with a spring).
In the final close-up shot of The Shadow, the webbing of his false eyebrows is visible.
Margo claims to have had her dress made by Adrian. While the dress does look like an Adrian-style design, he was under contract to only work for MGM until the early 1940s when he opened an independent store in Beverly Hills. It's not quite an anachronism for her to have an Adrian design in the late 1930s, since Macy's "Cinema Shop" was openly allowed to copy Adrian's dresses with MGM's implicit approval. She is likely bending the truth a bit that Adrian made it especially for her; it was more likely from Macy's.
The wire insulation on the bomb is modern day plastics not the correct cloth type for the day, and the nylon wire ties wouldn't be invented for another 30 years or so. The timer in the bomb used a display tube called the Nixie tube to show the digits of the countdown. The original Nixie tube was patented in 1938, but it looked like a small TV screen on which the numbers were displayed. The tube design used in the bomb was not introduced until 1962.
The movie is set in the 1930s. The two "longslide" 1911 pistols wielded by The Shadow were not developed until decades later.
At the start of the fight on the balcony, when the Shadow lifts one of Khan's men by his helmet, you can see the wires pulling the guy off his feet.
During the scene where the Shadow saves Dr. Roy Tam on the bridge you can make out the soundstage's EXIT sign to the right of the cityscape backdrop.
When Margo approaches the table Uncle Wainwright starts to introduce Lamont as though she has never met him, but Wainwright was in the restaurant when Lamont went over to Margo's table. He had to see them meet and leave together so she obviously knew him.
When Khan says Yinko, Farley immediately says, "the Shadow." There are two problems, first Farley does not know that Yinko is Lamont, second, he does not know that Lamont is the Shadow.
Dr. Lane cuts the wire on the bomb then a few seconds later when he goes to attach the ends the insulation is stripped back exposing the bare wires even though he never stripped the wire.
When Margo comes over to the table in the Cobalt Club, uncle Wainright introduces her to Lamont as if they were complete strangers, but during their previous dinner he saw Lamont go to her table, introduce himself, and leave with Margo, so knew they were not strangers and no introduction was necessary.
When Dr. Lane sees the hanging sphere for the first time, he says it is impressive work, but all he is looking at is a large metal sphere hanging from the ceiling, he cannot even see the timer, so there is no logical reason for him to make that exclamation.
Khan says that Genghis Khan conquered half the world. In fact, he conquered about one-sixth of the land area of the world, or one-third of the land area of Afro-Eurasia. If one counts only Eurasia, however, then the one-half figure is close.
The narration states that Lamont can never hide his shadow. Meaning his shadow is always visible even when he isn't. When Lamont sees one of Khan's warriors following him, he ducks into an alleyway and his shadow is seen on the wall next to him. When Lamont makes himself invisible, his shadow becomes invisible too although it should still be seen on the wall.
Dr. Tam report that an explosion will occur when the MOLECULAR bonds are broke. Atomic bombs are reactions within the NUCLEUS of the atom thus the name ATOMIC BOMB.
Lamont is able to read minds, sense people as he did when Khan was in his sanctuary, and control people through mental power, yet he has no idea Margo is in his house, even after she opens the door right in front of him.