The Mighty Pigs hockey team is a parody of the Anaheim Mighty Ducks hockey team.
Large sections of the episode are references to The Mighty Ducks. Milhouse being tied to the goals could be a reference to the goalie being tied to the goals in training to teach him to not be scared of being hit.
The school auditorium was named the "Butthead Memorial Auditorium", a reference to the series Beavis and Butthead.
The auditorium being named that, as well as Skinner's proverbially kicking himself afterwards upon making the announcement, was overall a reference to an incident in Greenwood, South Carolina, where various children voted to have their school named "Springfield Elementary School" after the eponymous location from The Simpsons as well as the resulting controversy. Ironically, in a press release by Fox News, Bart was stated to have said in response to the school's naming "At least it wasn't named Beavis & Butthead Elementary".
After the game, Homer gives Bart his turtle. After they hug, the turtle disappears.
When Bart hits Lisa with trash, the trash is shown. When Apu sees Bart hitting trash at Lisa, it is missing, so the trash appears to come from nowhere and Bart is shown hitting the ground.
When both Marge, Homer, and Maggie are shown rooting, the order from left to right is Marge, Homer, and Maggie. When Lisa and Bart make up, Marge is shown where Homer was.
When Krusty sings the National anthem badly, he forgets people aren't that bothered, even though forgetting the lyrics to the national anthem and/or singing it poorly has tarnished and/or ended the careers of many a singer.