Continuity: In the beginning of the episode Topanga and Cory are talking and Topanga tells Cory "Cory you're the one who broke up with me; don't you understand that?" This is not correct. In the 12th episode of the 3rd season, "The Grass is Always Greener" Topanga and Cory mutually decide to break up, with Topanga initiating the idea.
Goof: In 1957, Morgan is the only one who knows Cory, while his friends all think he is a new student.
It's 1957 and TL is doing the Twist.
The Twist didn't debut until 1960. However, since the episode is actually Cory's dream, it would make sense that there are anachronisms like this. It's not really 1957, it's Cory's perception of what 1957 would be like.
Mr. Feeny: Young lady, your hair is far to high; a clear violation of school policy and a hazard to all low-flying aircraft.
TL: If I see an ae-ro-plane, I'll duck, a'ight?
Cory (waking up from his dream): I'm not Brad Pitt! I'm not Brad Pitt!
Morgan: Well, duh.
Mr. Turner: How about you Shawnzie; what do you think life will be like in 40 years?
Shawnzie: I don't know. I don't know what I'm having for lunch.
Mr. Turner: That was this close to a shenanigan.
Cory: Topanga, there you are!
TL: Listen Archie! The name's T-L!
Cory: As in Topanga Lawrence.
TL: As in Tough Luck, for suckers who don't know better.
Topanga: Why are you looking at me like that?
Cory: I will always look at you like this.
Topanga: Well stop.
Topanga: Because your giving me the heebie-jeebies.
Mr. Feeny: I heard a scream.
Cory: Oh, Mr. Feeny, don't expect to get my paper on time.
Mr. Feeny: I never do.
Eric (in the 50's flashback): I've been accepted to both Harvard and Yale. You're my mentor. What's your advice?
Mr. Feeny: Oh you'd be a fine student at either institution.
Eric: Oh, undoubtedly I would, but they've asked me to teach.
Mr. Feeny: Well as you know, Mr Matthews, there are no females at Harvard.
Eric: Well I wouldn't anything to distract me from my intellectual pursuits. Harvard it is.
Cory (upon waking): Shawnzie?
Shawn (bewildered): Yessie?
Cory (waking from his dream): I'm not Brad Pitt! I'm not Brad Pitt!
Brad Pitt is a successful American actor,
Cory: Why are you guys talking like Boris & Natasha?
Boris & Natasha are fictional animated characters from the 1960s cartoons, Rocky and His Friends & The Bullwinkle Show.
Mr. Williams: I am clearly the hep cat.
Alex Désert is a founder and lead vocalist for the L.A. ska band, Hepcat.
The name, Shawnzie, is an obvious reference to the Happy Days character, Fonzie.
Topanga: Hey, Howdy Doody; we're dancing here!
Howdy Doody was a children's TV show from 1947 to 1960.
Wise Man: What do you want me to say? "Wax on, wax off"?
This is a reference to Pat Morita's other famous role; Kesuke Miyagi in, The Karate Kid.