In the film, Sigsbee Manderson (played by Orson Welles) mentions a performance of Shakespeare's "Othello" at the St. James Theatre in London in 1951, in which he disliked the leading actor's performance. This is an in-joke: Welles himself played Othello at the St. James in 1951, under his own direction. Peter Finch played Iago opposite him.
Orson Welles was struggling to complete his long-gestated film of "Othello" when his old friend Herbert Wilcox approached him to play the small but vital role of Sigsbee Manderson. Badly in need of money, Welles agreed, and wrote and directed his brief scenes. Wilcox claimed in his autobiography that he was so pleased by Welles' contribution that he actually paid him more money than Welles had asked.
The original novel was published in 1913, and there was a silent version of the same story made in Hollywood under the direction of Howard Hawks (Trent's Last Case (1929)). The original author, Edmund Clerihew Bentley, was a comic poet who disliked detective stories and conceived of the novel as a parody and an attack on the form. It is notable that the "brilliant amateur sleuth" who is the hero of the story manages to get everything wrong with his "clever" deductions and eventually only solves the mystery by accident. Despite this (Bentley was open about his intentions), the novel became a classic of the genre and most film and television adaptations, including this one, have played the story entirely straight, and not as the belittling joke Bentley intended.
Trent informs Marlow that he knows of his acting career at Oxford University, particularly his successful portrayal of Mercutio in Hamlet. Mercutio is a character in Romeo and Juliet, and Trent's education and career in the arts would mean that this was a deliberate error.