Aesop Amnesia: Sharpay suffers from this, having apparently seen the error of her ways at the end of the first film, only to become nasty again at the beginning of the sequel. Repeat for the second and third films.
Alpha Bitch: Sharpay was undoubtedly intended to be unsympathetic, due to the target audience. She comes across as more of a Lovable Alpha Bitch due to the way Ashley Tisdale portrays her, to the point where she became noticeably more humanized in the third film. She even got her own spinoff movie, Sharpay's Fabulous Adventure, where she went from Lovable Alpha Bitch to Fallen Princess.
Fun fact: Lucas Grabeel actually played a closeted gay on Veronica Mars, and he seemed straighter than Ryan.
All Guys Want Cheerleaders: Averted. There aren't any major cheerleader characters until #3, and even then, the only reference we get to a date with a cheerleader (in this case, Jason wanting to date Martha, the hip-hop loving nerd) is in a deleted scene.
Almost Kiss: Troy and Gabriella at the end of the first film before the Crowd Song, and then in the entire second movie.
Angry Black Man: Chad seems to exist largely for the purpose of angrily over-reacting to things.
Captain Obvious Aesop: "Be yourself and don't bully others". Quite an revolutionary concept, wouldn't you agree?
The Cheerleader: A mild version, but yet a scene in the first movie depicts cheerleaders as shallow, concerned only on having strong fingernails and guy's looks. (Or at least, depicts the perception Taylor has of them.)
Clingy Jealous Girl: Sharpay, who repeatedly thinks up plans to get Troy away from Gabriella and to become her boyfriend.
Continuity Nod: Is the only way to explain how Sharpay forgets the date of the Big Game in #3—it's an in-joke referencing her first line in #1.
Corrupt the Cutie: This was very nearly the entire plot of all three movies. #2 especially had Sharpay pulling out all the stops in an attempt to lure Troy away from his buddies, via "Look what I could do for you with all my nifty social connections."
E = MC Hammer: Of course Gabriella has to be able to do the ridiculously complicated equation on the board in her head. And then the teacher looks at the calculator for maybe two seconds before looking back up...
Exactly What It Says on the Tin: Sort of justified, however, in that it started out as a Disney TV movie that they didn't expect would be so popular. Had they any inkling, it probably would have been released to theaters.
Fake American: Oleysa Rulin (Kelsi) is originally from Russia.
Follow the Leader: The success of HSM led to Nickelodeon's blatant ripoff, Spectacular.
Foreign Remake: A Chinese one, with the involvement of Disney.
Grand Finale: Subverted. The end of the third movie seems to end the franchise for good, until you find out about the production of a spinoff movie starring Sharpay.
Heel Face Revolving Door: Sharpay and Ryan. They seem to become "good" at the end of the first film, then Sharpay is suddenly worse than before in the second film and Ryan just becomes good, then Ryan is back on Sharpay's side in the third film and they both go good again in the end anyway!
High School Dance: Subverted in the third movie: There is buildup—so much so that an entire song is devoted to it in the ''Senior Year musical...then Gabriella leaves early. Troy winds up spending Prom Night in California with her (as opposed to the movie's New Mexico setting), and the actual dance is confined to a single Imagine Spot (reprising the song "Can I Have This Dance"). The movie's actual climax is at the musical performance.
Hollywood Nerd: Every single character who's part of the "brainiac" table is this. Even Kelsi had to hidden under hats and glasses (which she doesn't even need), and she's still very attractive.
Honey Trap: Ryan, on Sharpay's orders, to Kelsi in the HSM 3.
Incestuous Casting: One character remarks that if Sharpay could find a way to play both Romeo and Juliet, Ryan would be out of a job, suggesting that the two have played romantic partners in the past.
Insane Troll Logic: "Look at this! That Gabriella girl just dumped her lunch on me! On purpose! It's all part of their plan to ruin our musical. And Troy and his basketball robots are obviously behind it! Why do you think they auditioned? After all the work you've put into this show... it just doesn't seem right!"
Nice Job Breaking It, Hero: Chad and Taylor's Zany Scheme to con Troy into confessing he doesn't care about the callbacks for the musical, and forcing Gabriella to watch while he does so. Their intent, ostensibly, is to get Troy to focus on the upcoming basketball game, Gabby on the scholastic competition. What ensues is nearly the termination of the One True Pairing.
Oh God, With The Verbing!: Jack Scott in the stage version. "A skaterdude was even seen mixing with the string section. Oh, the humanity... THE HUMANITY!"
Pair the Spares: Chaylor, Ryelsi, Zekepay, need I go on? Also takes place in Fan Preferred Pairing territory, where Troypay fans ship Gabriella with Ryan.
Passing the Torch: In a deleted scene to the third movie's new characters.
Serious Business: The passion and drama put into songs about things like the jock and the genius daring to try out for the musical reach ludicrous levels.
For example, see "Stick to the Status Quo". "No, no, NOOOOOOOOO!!!" about a basketball player being able to bake, a nerdy girl liking hip-hop, and a skater dude playing the cello indeed.
Terrible Interviewees Montage: The first film features one of these as a variety of terrible wanna-be actors try out for the school play. This is supposedly to show how talented Ryan and Sharpay are. However, as their competition are people who can't sing on key, people who forget their lines, people who are creepy, and people who lock up with stage fright, it actually creates the impression that the only reason they keep starring in school plays is because they are the only people who are vaguely competent at acting or singing.
Took a Level in Jerkass: Sharpay wasn't nice in the first movie, sure, but the second movie blew her negative qualities up to ridiculous proportions. She's back to a more reasonable level of "evil" in the third.
Twincest/Brother-Sister Incest: Referenced. Ryan and Sharpay don't mind performing love songs together, but then again, the key word is perform.
It also helps that the songs are rearranged to downplay the Squick factor; the love song "What I've Been Looking For" is sped up and fits a brother-sister bond nicely, and "Bop to the Top" is clearly addressed to the audience, not each other.
On the flip side, there's an awful lot of fanfiction out there that pair the two. A few explain it away as Ryan and Sharpay not actually being related, though most keep the relationship intact. Of course, many fics lean towards the angsty, tortured side.