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Spoilers (2)
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.
The poems in the film came from Ron Padgett, one of Jim Jarmusch's favorite contemporary poets, who agreed to write the poems for the film and who let Jarmusch use some of his pre-existing poems.
The conversation between the two construction workers in the bus was a real conversation Jarmusch overheard at a bar.
Co-stars Kara Hayward and Jared Gilman who both made their debuts in the Wes Anderson film, Moonrise Kingdom (2012), in which they played lovers.
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.
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.
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.
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.
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.
Spoilers
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.'