After the "Drum Boogie" sequence, the black jazz trumpet soloist Roy Eldridge disappears from the band.
In the car fleeing the nightclub: Sugarpuss O'Shea holds up Professor Potts's calling card to read it. Then the camera angle goes to a close-up of the card, with her fingers holding it around the four corners of the card, but the position of the fingers in the close-up shot do not at all match the way she is holding the card in the previous "long" shot.
When Potts declares himself "mystified" at the slang he has heard, his arm changes position between shots.
Sugarpuss notes that she "once watched her brother shave"; Joe Lilac (pretending to be Sugarpuss's father) notes that Sugarpuss was an only child.
When the gangsters visit the professors' house, they beckon to Sugarpuss to come out by the front door. After speaking to them, she is seen from inside the house returning via the front door, but when the camera moves outside, instead of seeing her on the steps that lead to the door of the brownstone, we see her emerging into a narrow alley, and the door has changed from the large, elaborate front door to a much smaller and plainer one.
A cop on the Washington bridge checkpoint says that a professor's driving license was issued in 1903. Driving licenses have existed in the USA only since 1910.
Joe wants to marry Sugarpuss who knows enough to get him convicted of murder because "a wife can't testify against her husband." In fact, the law prohibits forcing a wife to testify against her husband--ie, threatening her with prison, as can be done with other recalcitrant witnesses. A wife can indeed testify against her husband if she does it of her own free will. Further, Joe tries to force Sugarpuss into marriage with threats. A marriage performed under duress has no validity in law.
While Professor Oddly is reminiscing about his honeymoon in the Catskills, he refers to the Anemone Nemorosa plant. As a botanist, Professor Oddly would have been aware that the Anemone Nemorosa is a plant indigenous to Europe and the British Isles in particular. However, he never actually says that he saw the plant while on his honeymoon - simply that he regards the plant as a suitable metaphor for romance.
When Pastrami and Anderson shoot up the professors' work space, they each do so with revolvers. But they fire 8 times each before the camera goes to the housekeeper, the garbage man, and Oddly. When it goes to these three characters, and it is no longer possible to count each of their shots, the firing continues. Yet, at no time do these two gunmen reload their 6 shot revolvers after having fired more than 8 shots, and as many as another 10 shots total.
During the walk through central park, the camera shadow is visible on Professor Gurkakoff's torso.