Mr. Burns' attempt at bribing the Nuclear Regulatory Commission inspectors is a parody of Let's Make a Deal.
Mr. Burns' attempt to beat a member of Springfield University's Board of Directors with a baseball bat (which he barely notices, and which does no harm to him) is a reference to the 1987 Untouchables movie.
Bart scraping his nails across a chalk board is an obvious reference to Jaws (1975) in which Quint (Robert Shaw) does the same.
According to The Simpsons Guide to America's Favorite Family (and shown in this episode), Homer did attempt to apply to college in his youth. His high school guidance counselor told him he was a shoo-in for admission, and the only remaining step was to sign the application form. Homer was just about to sign when he looked out the window, saw a dog with a piece of ham, and ran outside to fight with the dog for it. He never made it back into the counselor's office to sign the application, and the frustrated counselor crumpled up the form and threw it in the trash. Homer, recalling the incident, said that "fate stood in the way" of his going to college.
According to The Way We Was, Homer was meeting a guidance counselor over a separate issue (how to get to know Marge), when the guidance counselor gives him a job brochure about the Springfield Nuclear Power Plant, remarking how it is one of the few remaining jobs that will not require a university education. During this episode, when Mr. Burns is being reprimanded by the regulators, one tells him "Homer Simpson's job requires college-level training in nuclear physics" and that Burns to get Homer qualified or else face sanctions.
Homer misspelling smart while burning his high school diploma was actually a result of Dan Castellaneta misreading his lines during recording. The producers loved it, and the rest is history.
When Mr. Burns tells Homer that he needs to go to college, Bart jokingly asks, "College? Barber or Clown?" Ironically, in a later episode, "Homie the Clown", Homer does end up going to a clown college.
This is the first time Homer gets a pet pig. The second time was in The Simpsons Movie. The main difference is that in this episode, he got it illegally, while in the movie, he got it legally.
During the bra bomb prank in School of Hard Knockers, the President is shown with five fingers instead of four while dancing as bras fall on his head.
When Homer sits on the couch, he has crossed legs, but later he has them side by side.
Homer refers to his "High School Diploma." He does not have one, he has a GED, as he went to night school to make up for his lack of a diploma. The document he sets fire to even clearly says "GED" on it. This was an intentional mistake on Homer's part.
In the dorm there is no ladder for the bunk bed, therefore the only way to get to the second or third bunk is to climb up the side, however the bed isn't screwed to the wall so it would fall over.