After gym signups, Lisa is shown wearing hockey gear, in reference to her previous stint as a goalie in "Lisa on Ice".
Homer tells Mel Brooks that his movie Young Frankenstein "scared the hell out of him." In actuality, Young Frankenstein (like most Mel Brooks movies) is a comedy, which explains Brooks's less than enthusiastic response to Homer's comment.
Homer mistakenly calls his classic "2000 Year Old Man" sketch "That 2000 Pound Man thing". Chief Wiggum later calls it "The $2000 Man".
In one scene at the ballet class, Bart is drinking Tab soda.
Bart's ballet teacher tells him to "leap like he's never been before" is a reference to the 1987 Simpson short, "Bart Jumps", where Homer tells him the same thing.
Patty and Selma's supervisor refers to Homer smoking at government building as "worse than Hitler." It is problematic on two fronts. One, the supervisor's analogy is an example of "Godwin's Law". Second, while he was a heavy smoker in his younger days, Adolf Hitler later finds smoking so distasteful, it was a reason why Nazi Germany led the first anti-tobacco movement in modern history.
This was a joke about then-recently enacted laws prohibiting smoking in workplaces.
In Latin America, the nickname "Ratboy", which Homer uses to refer to Bart at one point of the episode, has taken great fame on Internet, in large part, thanks to various YouTube videos, the phrase translated as "Niño Rata". Internet users use this phrase to refer to the inexperienced and immature video game fans between 7 to 12 years old who do not know how to lose, because if they lose, they scream, mainly fans of video games like Minecraft, Call of Duty, FIFA, Grand Theft Auto Online or Fortnite. It's also used to designate those YouTube users who do not provide opinions in the comments bar and only bother other users.
When left alone, Bart does an impression of ventriloquist Señor Wences to amuse himself.
Barney smokes a cigar without removing the wrapper and sees "Sgt. Pepper" (a reference to the album of The Beatles: "Sgt. Pepper's Lonely Hearts Club Band") growing out of Homer's back.