Adam Driver went to bus driving school for his role in the film. Production crew was arranging for Driver to get a bus license, and while they were trying to organize it, he on his own figured it out and was already in the school.
On Empire podcast #237, Adam Driver revealed that he underwent training and became a licensed bus driver. He wanted to be able to be on "auto pilot" while driving the bus. It also meant that the film could feature more authentic footage opening up the possibilities for a greater variety of camera shots. He was taught over a period of three months in Queens, New York City, passing the test one week before filming began.
The man rapping in the laundry is played by Method Man, a member of Wu Tang Clan. In Jarmusch's 'Ghost Dog' The RZA, also a member of Wu Tang Clan, has a cameo and he also wrote the score for the film. RZA along with GZA of the Clan also play a role in Jarmusch's Coffee and Cigarettes.
Laura mentions a dream in which she had twins. After this point & throughout the film, Paterson encounters twins on his way to work, in the bar, and twice on the bus. Also, the young poet appears to be a twin.
Laura says she has the same name as Petrarch's wife (Petrarch was an Italian scholar and poet of the 14th century). Actually, Laura was the muse of Petrarch; she never was his wife and they actually had limited contacts, if any.
Some of the books visible in Paterson's basement are "Walden" by Henry David Thoreau, "Lunch Poems" by Frank O'Hara, "Paterson" and "The Collected Earlier Poems" by William Carlos Williams, "The Fall" by Albert Camus, "The Walk" by Robert Walser, "Double Indemnity" by James M. Cain, "Bambi" by Felix Salten, "Alone and Not Alone" and "Great Balls of Fire" by Ron Padgett, "The Collected Poems" by Wallace Stevens, "On the Great Atlantic Rainway: Selected Poems 1950-1988" by Kenneth Koch, "The Call of the Wild/White Fang" by Jack London, "Collected Poems, 1956-1987" by John Ashbery, "The Soul of an Octopus: A Surprising Exploration into the Wonder of Consciousness" by Sy Montgomery, "In the Palm of Your Hand: A Poet's Portable Workshop" by Steve Kowit, "Selected Prose and Poetry" by Edgar Allan Poe, "Save the Last Dance for Satan" by Nick Tosches, "The New York Trilogy" by Paul Auster, "I Fought the Law: The Life and Strange Death of Bobby Fuller" by Miriam Linna & Randell Fuller, "Kill All Your Darlings: Pieces 1990-2005" by Luc Sante, "Gone Man Squared" by Royston Ellis, "This Planet is Doomed: The Science Fiction Poetry of Sun Ra" and "Prophetika Book One" by Sun Ra, "The Baltimore Atrocities" by John Dermot Woods, "Sweets and Other Stories" by Andre Williams, "Benzedrine Highway" by Charles Plymell, "I Remember" and "The Nancy Book" by Joe Brainard, "Monk" by Laurent de Wilde, "Lawrence in Arabia: War, Deceit, Imperial Folly, and the Making of the Modern Middle East" by Scott Anderson, "Consider the Lobster and Other Essays" and "Infinite Jest" by David Foster Wallace, "The Flowers of Evil" by Charles Baudelaire, "Up Above the World" by Paul Bowles, and "Amerika" by Franz Kafka.
Characters throughout the movie remark on the coincidence of Paterson (Adam Driver) having the same name as the town in which he lives (Paterson, New Jersey). Paterson, a bus driver, is played by an actor who is coincidentally named "Driver."
The trivia items below may give away important plot points.
There was no reference to twins in the original script. In a Q&A at the Landmark Sunshine Cinema in New York Jim Jarmusch revealed the serendipity of the twins in the film. At the beginning of filming he realized one of the children extras on a bus scene was, in fact, a twin, when her mother and sister came to pick her up. He told the mother to bring both sisters for filming the next day, and started to include twins throughout the rest of the shoot. He then added Laura's dialogue about her dream with twins. To him, the twins (as well as the dog-jacking foreshadowing) in the film are "anti-significant." One might think at the end of the film Laura would be revealed to be pregnant with twins or that Marvin would be stolen, but neither occur.
After the notebook is destroyed, Paterson mentions that they were only words written on water. This might be a reference to the English poet John Keats, whose gravestone epitaph reads 'Here lies One whose Name was writ in Water.'
The last poem Paterson writes with the line; "Would you rather be a fish", is from the song 'Swinging On A Star' (Burke/Van Heusen): "A fish won't do anything but swim in a brook / He can't write his name or read a book / And to fool the people is his only thought / Though he slippery - he still gets caught / But then if that sort of life is what you wish / You may grow up to be a fish."