There isn't much of a story line to speak of, and to call it a character study would be generous, since we don't really get that deeply inside the titular character. Even so, all the things that make a Coen Brothers film so enjoyable are here. The dialogue is smart and pithy, individual scenes are immaculately constructed and completely engrossing, and the peripheral characters are hugely entertaining.
Llewyn Davis isn't the most sympathetic of protagonists. He's a sullen young man who sponges off the few friends he has, sleeping on their couches, bumming their cigarettes and eating their food. If they are musical peers he's miserly with praise towards them, if they're outside the industry, they're only good for their money. The most charitable thing that could be said for him is the loss of his singing partner has obviously left a hole in his life.
Carey Mulligan and Justin Timberlake play fellow folk singers Jean and Jim.
Curiously, the one area in which Davis shows a sense of responsibility is his constant scrambling to look after his friend's runaway cat. The cat actually deserves its own Oscar for best performance by an animal, showing an uncanny array of emotions.
While Davis' dogged conviction not to be 'careerist' means his journey throughout the film seems an aimless one, and he is so dour in character, the film is still full of humour thanks to the rich milieu that the Coen Brothers have created. Davis' sleazy, old agent and his crabby (and just as old) secretary are a comedic delight, as is John Goodman as a verbose jazz musician who has little respect for folk music.
If you feel the same way as Goodman's character, it may affect how much you enjoy the film. There are frequent musical performances throughout with various cast members lending their voices to folk standards (plus one original song).