At 16:53, between camera cuts, Louie's cigar jump from his left hand to his right hand.
Around 19:14 the gang is watching the changing of the guard at Buckingham Palace. Benny Bartlett's character "Butch" is standing next to Slip Mahoney.
When they cut to a closer shot, David Gorcey's character "Chuck" is now next to Slip.
The attorney who informs Sach about Lord Walsingham uses the American pronunciation of Walsingham (WALL-sing-ham), even though he introduces himself as a solicitor and has a British accent. Later we learn how differently "Walsingham" is pronounced in England (WIZ-um), which is the pronunciation the British solicitor should have used. That sounds all well and good except that it's Slip who pronounces "Walsingham." J. Allison Higby, the solicitor, never utters it in the entire film.
The dungeon has been out of use for some time, presumable decades, yet during the sword fight there's a lit cauldron heating up a branding iron, a vat full of water, and sconces alight with flame.
The shadow of a boom microphone is visible on the wall above and behind Lady Marcia and Sach as she tries to lure him away and get his secret.