Factual error: The beer bottle behind John Goodman in the picnic scene is a modern day Budweiser bottle.
Factual error: The film takes place in the Thirties. The song "You are My Sunshine" is featured, but was not recorded by Jimmie Davis (its composer) until 1940. [Acknowledged by the directors].
Continuity mistake: When they first meet George Nelson, the left back seat window is half way open. A few seconds after they get into his car, Delmar gives the machine gun to Nelson through the left back seat's completely open window (the camera shows the guys all the time so we can see that Delmar didn't lower the window when he got in the car).
When Everett asks the hobos on the train if any of them are "smithys", look closely and you'll see that they're sitting on big bags of Pappy O'Daniel flour. Pappy is a major character later in the film.
George Clooney (Everett) was going to sing the song "Man of Constant Sorrow" for the film but it turned out that he was rather awful, so he ended up lip-synching the songs instead. He said "I'm not my aunt [referring to the late singer/actress Rosemary Clooney, best known for her role in "A White Christmas" (1954)]. I decided it would be easier to just do a passionate lip-sync." He was so nervous that the tapes of his singing would get out that he returned to the studio to make sure all of the evidence had been erased. The musical director of the film confirmed this but said "George is a very good singer but that style of music is very difficult and one almost has to grow up singing it in order to sing it convincingly. George did a really good version of the tune but it wasn't as good as he wanted."
Just like the classical Odysseus, Everett openly defies God, and has constant troubles because of it. The second he repents and asks for God's help he is saved from death and everything turns out OK.