These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Big Lipped Alligator Moment: The film opens up with the story of a Jewish couple and a rabbi who may or may not be a dybbuk (a reanimated spirit in Judaism). This is not mentioned again afterwards; it's supposed to give a demonstration of the film's conflicts (faith vs. reason, etc.).
You could count most of the film as one of these: it's a kaleidoscope of angst-ridden Jewish whimsy. It's the accumulative effect of the recurring verbal symbolism (including the first scene) that gives the film depth. At a shallow glance it just seems like one utterly random scene after another, doubly so for the dream sequences.
A common interpretation of this movie is that it is, at least in part, a modern retelling of the Book of Job from the Old Testament. How does Job end? God appears to him in the form of a whirlwind. Doubles as Viewers Are Geniuses.
As discussed here, excluding the prologue, the first and second half of the film are reflections of one another.
Despite the absence of random acts of Ax-Crazy violence, this is still arguably one of the scariest of the Coens' films due to its thematic implications: bad things happening to good people for no reason, and the suggestion of a God who is both merciless and utterly incomprehensible.
To a lesser extent, the prologue. Though nothing openly frightening happens, it's left a lot of viewers very uneasy.
One-Scene Wonder: The extras casting in this film is absolutely pitch-perfect, and most of the minor characters only flit in for a scene and then disappear, so there are a lot of these.