1. Hail, Caesar!
Related Thing: Why Do People in Old Movies Talk Weird? & tracing the roots of the homage in Hail, Caesar!
2. Raising Arizona
Related Thing: I see Raising Arizona as following the classic mythical western template. There's no better example of that than the most iconic spaghetti western of all time, The Good, The Bad, and the Ugly. Sergio Leone intended to create a satire. He accidentally elevated that which he intended to critique. He was simply too good at it.
(The most popular theory to emerge states that the three main characters of The Good, The Bad, and The Ugly are Gods who have descended in the disguise of humanity to wage spiritual battle on Earth.)
The Coens have an innate understanding of how tight scriptwriting and intense focus on small details can amplify a seemingly run-of-the-mill story. A focus on simplicity of plot and clarity of character transforms a story into an artifact. Something audiences can read meaning into, as so many have done with Raising Arizona. (Many people posit that the entire film is Hi's dream sequence or slow death, populated by people he interacts with while serving time.)
In other words, good writing consistently stokes fan theory culture.
3. The Hudsucker Proxy
Related Thing: Pee Wee's Lament and relevant information about why I think Jennifer Jason Leigh's character is an overt homage to Rosalind Russell in His Girl Friday.