These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Big Lipped Alligator Moment: In the second movie Sharpay tries to get Troy to perform Humuhumunukunukuapua'a with her. What is Humuhumunukunukuapua'a? An extravagant Hawaiian themed number about a pineapple princess and a fish prince falling in love. Characteristic of Sharpay, this is a big production number with three backup dancers and Ryan narrating, providing sound effects/fog, and playing the ocean and fish. Much Narm is involved. It makes just as much sense in context and is never brought up again or acknowledged in the plot. No wonder it was cut from the Disney Channel airings.
Broken Aesop: Between #1 and #3 vs #2. The moral of the first film is essentially 'Support other's dreams rather than tearing them down' (The wildcats accepting Troy and Gabriella singing). The second film switches to 'Your friend's pressuring is more important than your dreams.' (Troy trying to win a scholarship). Then the third film jumps back onto the 'support each other' wagon. (Troy doing basketball and singing, Gabriella going to Stanford, Kelsi wanting to do the show together). If you're confused, fair enough.
Perhaps it's not to sacrifice your friends for the future or that the means don't justify the ends.
High School Musical 2. Some saw it as a step in the right direction but many people pick up on a lot of issues, in particular the Informed Wrongness of Troy's decisions (how dare you miss a baseball game to work for a scholarship!), the Family-Unfriendly Aesop of the wildcats reactions (focusing on the future is bad), Sharpay taking a ridiculous level in jerkass, and most of all it's not even set in high school!
Sharpay's Fabulous Adventure. It's not a musical, and even factoring that and it's spin-off status in it has a completely different "feel" to the other movies. And those are the minor points - it forgets that Sharpay was supposed to go to the University of Albuquerque after graduation and that Ryan and Kelsi went to New York (they had to rush just to get Lucas Gabreel into the movie at all, so much that his scene is only in the broadcast version).
Designated Villain: Some people feel this way about Sharpay and Ryan, at least in the first movie. They have been doing all this musical extravaganza for most of their life and then in waltzes two people who have never actually sung seriously a day in their life and they easily got shooed in to get the parts. Although they can be pretty mean at times (especially Sharpay) you can at least understand why they would be pissed off.
However, they never had any real competition before so perhaps it's more that they weren't used to having to prove it.
Averted in the second film where Sharpay is a lot more mean-spirited, has no sympathetic justification for stealing Troy away from Gabriella, and treats the rest of the characters like dirt. The third film gives her more understandable motives again, as all she really wants is a decent chance of impressing Julliard, even if she goes overboard trying to get there.
Double Standard: Sharpay's pursuit of Troy in the second film comes across as sexual harassment and blackmail but she's Easily Forgiven by the end and her more extreme moves are Played for Laughs. Troy is clearly uncomfortable and nervous around her, but is still forced to shoulder most of the blame, even though she's actively stalking him and his job means he can't avoid her.
Ear Worm: Every Song. Especially in the first movie.
The second film has several, the worst one being that hanging out with your high school friends is more important than your whole future. Troy is portrayed as the bad guy for missing a few ball games because he's trying to get a scholarship and earn money for college. He's redeemed when he throws away chances of impressing the College board so the other wildcats will be friends with him again.
Fanon: Gabriella's Mom is nameless in the films, but is usually Isabella or Maria in fanfic.
Fan-Preferred Couple: While Troy and Gabriella keep most of the fandom support, a lot of the secondary couples get played with. In particular Chad/Sharpay and Chad/Ryan are often more popular than the canon Chad/Taylor.
Freeze-Frame Bonus: Miley Cyrus is among the dancers in the closing number of High School Musical 2. She gets a very brief closeup towards the end.
Ho Yay: Everything about the "I Don't Dance" sequence in HSM 2. The next time we see Chad and Ryan, they've switched clothes.
Gabriella is a gorgeous and brilliant new girl in school who immediately attracts the big man on campus and lands the leading role in a school play because of her stellar singing ability. To make things worse, while the first film at least portrays her as a Shrinking Violet with terrible stage fright who has no friends and no apparent hobbies outside of studying (until she tries singing), the second and third present her as super-popular and trained in various skills (life guarding, first aid, etc).
Troy falls into this too: Very attractive, great at almost every sport, so good at singing/theater he's put up for a scholarship at Julliard with zero training and the most popular guy in school. However both of them are somewhat saved by being a Broken Ace, not happy with the pressure on them and aware their popularity vanishes the moment they fail and let people down.
Surprisingly Improved Sequel: Between a better budget, more production time and the adorable chemistry Troy and Gabriella have developed through Zac and Vanessa's Romance on the Set, Senior Year is probably the best of the three films. Not to mention Troy and Gaby's relationship gets some depth, as per their confiding in each other about their fears for the future.
Each sequel got better in terms of budget and production, with the first one looking much less cinematic than the third.
Special Effect Failure: The second movie, during the song "Bet on It", has Troy looking into the water of a pond, showing his reflection. Ignoring the obvious CGI effect, the reflection itself was not mirrored.
Off-screen, but in the first movie a character says brother and sister Ryan and Sharpay played the title roles of Romeo and Juliet.
The fact that the theater director seem to have no issue with a brother and sister playing romantic interests.
Strangled by the Red String: In a badly-made excuse to stop the rumors that the character is gay, Ryan was paired up with Kelsi in the third movie. They are good together, if you ignore the fact Kelsi was paired with Jason in the first two movies.
Vanessa Anne Hudgens seemed to be contractually obligated to give the audience one overly emotional break-up ballad.
Zac Efron became obligated to supply one token emo, Narmy song per movie, famously including "Bet On It" consisting of Zac Efron interpretive-dancing angstily through the golf course in what's supposed to be a serious moment.
NO, NO, NO NOOOOOOOOOOOOOOOOOOOOOOOOOOOOO. STICK TO THE STUFF YOU KNOOOOOOOOOOOOOOOW.
The Woobie: Gabriella's backstory: It's implied her father has passed away and her mom's constantly-moving job meant she never settled anywhere. Plus she struggles to make friends because of being labelled the 'freaky math girl' wherever she goes.
Troy in the third film. Everyone around him is pressuring him to do different things, he has no idea what to do with his life, and his girlfriend - who he admits was the one person who doesn't see him as the perfect basketball boy - is leaving.