Randall 'Tex' Cobb was not familiar with riding motorcycles prior to filming. While shooting the scene where he rides up to inspect the hole where Evelle and Gale had escaped from prison, he actually crashed into the hole on one of the takes.
The prison counselor at the beginning of the film wears a Chai on his necklace. Chai is a Hebrew word and symbol that means "life," and is pronounced as if you were saying "hi" in English. Nicolas Cage's character "H.I." is called "Hi" throughout the film.
The news article H.I. reads early in the film about the Arizona Quints contains the following text: "Their father is unpainted furniture tycoon Nathan Arizona, who is reportedly pondering a run for Congress in the 4th district. Pete Peterson, Republican incumbent in the 4th, dismissed the birth of the quints as 'a cheap publicity stunt' in a news conference Thursday. He characterized Nathan Arizona as an 'unprincipled media hog and a loud hectoring nitwit', but conceded that 'Trey Wilson', the actor portraying him, as a 'very nice fellow with a distasteful job to do.'" [sic]
Although the letters POE and OPE are shown reflected in the mirror in reverse, the writing on the "We aim to please..." sign on the same wall is not reversed. Later, when the Biker breaks down the door, you can see that the sign is actually printed in reverse lettering.
When Ed tells H.I. not to cuss around Nathan, he responds with "He don't know a cuss word from Shinola". In the American South, to "not know shit from Shinola" (an old brand of shoe polish) means to be incredibly clueless.
The cigar-smoking bird tattoo was originally the logo of Clay Smith Cams in 1950s, a company making high performance engine parts. The logo, with the trademark clenched cigar, represents Smith himself and is known as "Mr. Horsepower". Smith closed the business in the 1960s and the logo was adopted by what is now Tenneco for their Thrush muffler line.
There really is a Farmers & Mechanics Bank (though note that in the film it's Farmers and Mechanics Bank, with "and" instead of an ampersand) in Galesburg, Illinois. But there was a bank of the same name (relationship to the Illinois bank unknown) in the Coen Brothers' home town of Minneapolis. The bank no longer operates in the Twin Cities, but its 1942 building is considered historically significant and is now a hotel.
At the end of the sequence where H.I.'s dream of the lone biker forms into the reality of Mrs Arizona discovering she is one baby short, the camera skims along the ground and jumps a child's bike, a car and then a fountain, before shooting up the ladder into the bedroom. This sequence seems to be a reference to the condensed career of motorcycle daredevil Evel Knievel, who jumped bikes, cars, buses and then spectacularly crashed after jumping the fountain at Caesar's Palace, Las Vegas.
Both O.P.E. and P.O.E. are spray painted on the door of the gas station rest room. This is a reference to Dr. Strangelove, as the recall code is O.P.E. and General Ripper is obsessed with P.O.E. (as it stands for both "Peace on Earth" and "Purity of Essence").
While many reviews refer to Randall 'Tex' Cobb's character Leonard Smalls as a "Harley-riding biker," the motorcycle in the movie is in fact a Honda Shadow that has been subjected to the "Rat Bike" treatment along with some extra flame-emitting plumbing. The Shadow is slightly smaller than a Harley Big Twin, which would give Cobb's character a bit more stature than he already has. One must also assume that since the bike is trashed near the end of the movie, it saves the production company a few coin.