Like the movie, every episode opens with a (false) declaration that it's based on a true story with the names changed.
The overall look of Season 3's cinematography is designed to invoke the soft, dreamy look of Inside Llewyn Davis.
"The Law of Vacant Places" starts with a tracking POV shot from inside a microphone and the story proper begins with a flashback scene that takes place in a faraway place that's only related to the main story thematically, as does A Serious Man.
Ennis Stussey operates a Red Owl grocery store, which is also seen in A Serious Man when Dr. Sussman goes to a Red Owl in the "Goy's Teeth." The chain of supermarkets was headquartered in Hopkins, Minnesota, but the brand was bought out in 1989.
Thaddeus Mobley's story in '70s Hollywood mirror that of Barton Fink. To drive the reference further, we got a shot of Gloria sitting on a beach, just like Fink. Additionally, Fred Melamed, who plays producer Howard Zimmerman, played Sy Ableman in A Serious Man.
Early on, Emmit and Sy make vague references to "Stan Grossman", the financial advisor of Wade Gustafson in the film Fargo.
The motel in the '70s flashback bears a resemblance to the Motor Motel in Sioux Falls, featured prominently in the last two episodes. There are also motels in Season One, the original film, and in A Serious Man.
In "The Narrow Escape Problem," Sy corrects Emmit on the difference of remains and cremains. Cremains is the preferred nomenclature for the ashes of the departed.
In "The Lord of No Mercy" Varga mentions Lenin and Walrus, like the conversation between Walter, Donny and the Dude in The Big Lebowski.
In "The Law of Inevitability," the opening scene where Nikki tries (unsuccessfully) to escape out the bathroom window of her hotel room is an imitation of Jerry Lundegaard's failed escape at the end of the original Fargo.
In Nikki's death scene, the movie's theme music is playing as the camera pans over her body and the trooper's body.
The Season 3 finale's title "Somebody to Love" is also the name of a Jefferson Airplane song that has a prominent role in A Serious Man.
Varga's thugs commit murders around a police vehicle on the side of the road. When a man and woman pass by and gawk at them before driving away, the most dangerous thug gives chase and executes them. All of this echoes the famous scene in Fargo.
Paul Marrane, an apparently supernatural entity who appears beside several characters at bars (one of which is located in a bowling alley), is a reference to The Stranger from The Big Lebowski.
Varga brazenly dismisses an (accurate) accusation presented to him as "mere surmise", and the person he's talking to has trouble parsing the phrase. There's a very similar exchange in A Serious Man.
Maurice tries to throw a joint out the window, but doesn't realize it's closed and it lands on his lap a laThe Big Lebowski.
In the season finale, Nikki assumes Varga has underwear in the briefcase instead of the ransom money. This was a major plot point in The Big Lebowski.
At one point, Gloria wonders if she does not exist. The theory sounds familiar to the theory posited in The Man Who Wasn't There (2001).
Nikki and Ray's relationship as a law-enforcement officer and a former criminal on parole is similar to Raising Arizona. There is even a scene where the two characters meet during a mugshot, which s nearly identical to a scene from that movie.
The ageing Zimmerman is a reference to the boss of the law firm from Intolerable Cruelty.
Others:
Emmit's contact is the non-existent Rick Ehrmantraut, whose surname is rare enough that it's clearly a shout-out to Mike Ehrmantraut of Breaking Bad and Better Call Saul.
Another Breaking Bad reference is when Varga finds a man to cop to the "Stussy Murders" in order to nullify Emmit's confession. Dialogue implies that this is similar to a plot point from Saul Goodman's intro episode where Saul hires a patsy who earns his living by taking the fall for people and going in and out of prison.
Paul Marrane (played by Ray Wise) is a mysterious man with apparently supernatural knowledge who makes a number of references to Judaism and the Hebrew Bible. His name is taken from the 17th-century French novel Letters Writ by a Turkish Spy, in which the name is used as an alias by the Wandering Jew.
The casting of Ray Wise as Paul Marrane may indicate a stealthy nod to Twin Peaks, as Wise's character in that film is also shown to be a dweller in a liminal space (the Red Room/Black Lodge). When Nikki alludes to their surroundings being a bowling alley, Marrane says "Is that how you see it?", implying that it takes different forms...
David Thewlis' character V.M. Varga philosophises aggressively about the exploitation of capitalism and wealth while appearing to delight in his belief in his own intellectual superiority over others. Varga shares these similarities with the protagonist of the 1993 Mike Leigh film Naked (also played by David Thewlis), who is a highly intelligent motormouth and a predator (of another sort) to boot.
The "misadventure by major appliance" mirrors one of the murders in the John Waters film Serial Mom.
The character of VM Varga bears a lot of similarities with the character Albert Spica, played by Michael Gambon in The Cook The Thief His Wife and Her Lover. Both characters are vocally sexist and anti-Semitic, both characters have a monstrosity represented by food, both characters have their thugs violently force someone to eat something disgusting, and both characters are played with similar English accents.
The final scene is extremely reminiscent of a scene near the end of Lord Of War where Nicolas Cage makes a very similar speech to his lawful captor about how their superior officer is about to walk in and release him.
The final scene also has VM Varga's dialogue referencing a famous quote by Ayn Rand about the "objective" usefulness of cats, which is often used as evidence of her hypocrisy.
