Matt Groening's name appears first in the list of gay people. In fact, the entire list is made up of crew members who worked on the show, all of whom had to sign waivers in order to have their names on that list.
When the comet flies over the Springfield city zoo you can see penguins flying away from the comet, a parody since they can't fly. It could also be a reference to the novel "The Hitchhiker's Guide to the Galaxy", where dolphins fly off Earth prior to its' destruction.
Homer tries to comfort his family by telling them that the comet will burn up in the atmosphere and whatever is left will be no bigger than a Chihuahua's head. When Bart eventually picks up the little piece that remains of the comet, a Chihuahua is standing next to him; a clear reference to Homer's earlier remark. Implying, he has some knowledge about rocks or asteroids. This was even referenced in the ending, where Lisa notes that Homer was right all along, and everyone in the family, Homer included, being terrified at the prospect.
On a related note, said Chihuahua, Taquito, appears in a few more episodes before gaining a very major role in the episode Dogtown as the main antagonist.
In the scene where Principal Skinner says "No" three times over aggravating situations, the last of them is over a freshly delivered newspaper that featured the headline "Prez Sez: School is for Losers!" At the time the episode aired, Bill Clinton had been president.
The numbers that Principal Skinner reads off while observing through the telescope ("6 hours 19 minutes Right ascension, 14 degrees 22 minutes Declination") are actual astronomical coordinates. Principal Skinner is observing in the constellation Orion, just to the east of the star Xi Orionis (about 2/3 of the way between the stars Betelgeuse and Alhena), slowly moving northwards until he reaches 6 hours 19 minutes Right Ascension, 14 degrees 59 minutes Declination, a distance of 37 arcminutes, about the apparent width of the full moon. Bart finds his comet at 4 hours 12 minutes Right Ascension, 8 degrees 7 minutes Declination, in the constellation Taurus, just southwest of the star Mu Tauri.
Everyone getting outraged and wanting to come into Ned’s shelter is a reference to "The Twilight Zone" episode, "The Shelter".
The "Three wise men" constellation is representing the Three Stooges' faces.
For a couple of seconds when Homer tries to get the shelter door shut when everyone gets in, Waldo from "Where's Waldo" appears rather obtrusively on the left half of the screen. In subsequent views of that shot, Waldo is replaced with Mr. Spock of Star Trek.
Skinner curses Pierre Jules César Janssen for having invented helium, even though helium is a naturally-occurring element and would have to be discovered. On top of that, Janssen didn't discover helium by himself (also a goof)
Buttzilla, the other name for the weather balloon prank by Bart, is a play on to the name of the Japanese monster known as Godzilla.
The siren raised by the observatory to warn Springfield about the comet, is the same as the air-raid siren to warn British people about Nazi bombings during the Blitz.
When the crowd joins Flanders on the hill during the "Que Sera Sera" song, Apu appears yellow.
Barney's hair is blue in one scene.
It was shown in the previous episode "Whacking Day" that Springfield Elementary School had an old fallout shelter in the basement. At least Seymour Skinner, Groundskeeper Willie and other Springfield Elementary staff members could have taken cover from the comet in there. Of course knowing the professionalism and craftsmanship of most Springfieldians it's possible the bomb shelter wasn't guaranteed to be totally safe.
When everyone is in Ned's shelter the woman in the top left hand corner briefly has two pupils in each eye.
When everyone is in Ned's shelter, in some takes, Jimbo Jones' hand appears white, mingling with Horatio McCallister's cap, in others it appears yellow.
When Bart first discovers the comet through the telescope, it is traveling in a westward direction. But when Bart shows the other kids the comet the following day, the comet is suddenly heading right towards Springfield and appearing to come right toward the camera in each shot. Even when seen from Flanders's backyard, the comet is moving straight forward towards the camera, but when the burnt-up comet bounces against the shelter, it instead comes in from the right side of the screen. Comets are unable to change direction in a short course of time in real life.
Comets are very cold and icy, and thus would not be continuously making a flaming sound like the Bart Simpson Comet did.