Angst? What Angst?: The preteen to teenage main characters aren't nearly as psychologically affected by always having to fight monsters that are trying to kill them so often as they probably would be in real life. At the very least, there's no sign of PTSD. However, given that they have to do this for years on end, it could be safe to assume they just get desensitized by it, or as demigods are more geared to handle such things.
Counterpart Comparison: While he's different enough to not be a full-blown Expy, Percy has a few similarities to Harry Potter. Both are supernatural heroes that were Raised By Muggles, didn't know about their abilities until the events of the series started, one of their caretakers is an abusiveFat Bastard, and they both go to a special educational facility with other beings like him.
Crack Pairing: Tons. Rachel/Nico, Thalia/Nico, Percy/Clarisse, Rachel/Luke, Apollo/Hermesnote Which is actually technically possible in the source material, especially given the fact that both are involved in same-sex relationships with mortals and other gods, etc.
Nico di Angelo gets the most love in this category, going from cute, annoying kid to an emo badass over the course of the series.
Fandom still showers attention on his sister, Bianca di Angelo, too. Never mind that she dies in her introductory book. Many fanfics explore either her life after death or her life in a "What if?" scenario should she have lived.
The twins, Artemis and Apollo, also get a lot of love, despite their very minor appearances. Quite a few fanfics are dedicated to them.
Similar to the above, the Hunters of Artemis get a lot of attention in fanfic, especially notable considering that very few members are named. Artemis's lieutenant, Zoë Nightshade, is especially popular.
Calypso has quite the following, and avoids the Die for Our Ship treatment that Rachel first suffered. Many rejoiced when she returned in the sequel series.
"Funny Aneurysm" Moment: Percy and Nico's first meeting is a pretty funny scene where Percy saves Nico's life and Nico proceeds to talk his ear off about mythology. It's significantly harsher after The Heroes of Olympus showed a glimpse of Nico's POV in that scene —basically, he saw Percy as a mythical hero come to life, which only made his crush on Percy and Percy's failure to save Bianca that much harder for him to accept.
Genius Bonus: It might or might not be intentional, but the mention of a statue of Susan B. Anthony strangling Frederick Douglass in The Last Olympian might be a roundabout reference to how the two civil rights leaders, formerly steadfast allies, became divided over the issue of the Fourteenth Amendment allowing black males to vote, but not women of any race (and to add insult to injury, it was the first Amendment that explicitly mentioned male and only male citizens).
Growing the Beard: Most agree that The Titan's Curse was where the series really began to show its quality.
Percy catching Annabeth when she nearly falls off Olympus becomes cringe worthy after the events of the third The Heroes of Olympus book, Mark of Athena, where he catches her again while she's being dragged by Arachne's threads into Tartarus and this time around, he chooses to fall with her.
Promise (which seems to be the Arc Words in Luke's life), after reading The Diary of Luke Castellan.
The tense relationship between Nico and Percy becomes downright depressing after The House of Hades, where it's revealed Nico was hiding his true feelings about Percy.
It's easy to miss it, but in his introduction scene, Mr. D quips how his father (Zeus) likes punishing him. This is bad enough but it becomes ten times worse when you read the Trials of Apollo.
Hell, when Luke gives him the enchanted sneakers, Percy admits the gesture made him blush almost as much as Annabeth. Who he then jibes for being obvious about her crush on Luke.
Percy and Nico share a bit in The Last Olympian. And then in later books, it turns out that Nico actually has/had feelings for Percy.
Travis and Connor Stoll (from the Hermes cabin). Percy and Tyson, too, but the Stolls are almost never without each other. Of course, the Stolls are also full siblings, which makes it kinda squicky.
Hype Backlash: Readers who never grew up with the series and curiously read it because of the popularity often met with disappointment.
Inferred Holocaust: In the first book when Percy, Annabeth, and Grover face Hades, the god of the dead becomes so enraged that he causes not one but two earthquakes, and Percy remarks that it will not have been a peaceful night in L.A. When the kids resurface, Los Angeles is burning. No casualty numbers are given, but it could qualify as a "Funny Aneurysm" Moment, as Hades had just finished complaining about how the Underworld was overcrowded and didn't need more subjects. Kinda ironic.
Manhattan sounds pretty bad by the end of the last book. Everyone had fallen asleep, which probably includes people who were cooking things, people who were in the hospital getting surgery, etc. And Fifth Avenue is a disaster.
I Knew It!: Who is the lightning thief? Could it possibly be Luke, the swordsman who is a son of the god of thieves, and has a scar running across his face?
Les Yay: Silena and Clarisse, Annabeth and Thalia, Zoë and Bianca. And just... the Hunters as a whole.
Clarisse and Silena in particular considering they play out the relationship between Achilles and Patroclus to its tragic conclusion. And we all know about them.
Rewatch Bonus: For those that read the sequel series, every single scene regarding Nico and Percy takes on a new meaning once you learn Nico had a crush on Percy since the day he was rescued.
Sailor Earth: With the sheer number of Greek Gods, major and minor, with dozens or hundreds of unclaimed children with many of them fighting on the side of Kronos it is very easy to begin imagining any number of demigod characters. This got taken even further with the advent of the gods' Roman counterparts, creating even more possibilities.
The Untwist: The fulfilling of the prophecy in The Titan's Curse It says "One will be lost" and everybody assumes it means someone will die. But it never outright states dying, so with the usual Prophecy Twist you could expect Bianca to be lost and later found. It turns out she gets Killed Off for Real.
The Woobie: Everyone not named Gabe, Ares, or Kronos:
Luke's mother, May Castellan, also deserves a special mention here.
In The Last Olympian, the narration actually says that Hermes "looked like he needed a hug." If that's not baiting woobiedom, nothing is.
MRS. JACKSON. Ridiculously nice, sweet, and caring, and yet gets get stuck with an abusive slob of a husband and serious financial problems. Luckily, things get much better.
Even Dionysus has a little of this after he revealed his relationship with Ariadne to Percy in the third book, and after one of his sons dies in the fourth book. And then you get to his chapter in Percy Jackson's Greek Gods...
Tropes which only apply to the film:
Angst? What Angst?: Percy seems to handle the sudden death / disappearance of his mother very well...
Anti-Climax Boss: Kronos in Sea of Monsters, possibly intentional because the rest of the books might not be adapted to film.
Badass Decay: Although her actual status as a badass is somewhat disputed (see Broken Base below), Annabeth still does a lot of things in the first movie and is actually helpful in several situations. Come the sequel and she does absolutely squat anymore despite tagging along as a main character for the whole film. Her position as the abrasive Action Girl has been taken over by Clarisse while Annabeth has become little more than a constant Damsel in Distress.
Whether or not Annabeth counts as a Faux Action Girl. Those who claim she is point to the fact that she has to be saved from Medusa by Percy, doesn't figure out the Lotus hallucination until after they're outside it and then does nothing in the climax - despite claiming to have lots of experience with battle strategy at camp. Those who say she isn't point to her beating Percy in their first duel (and he needs a power boost from water to match her), being the one to use Car Fu against Medusa, successfully shooting some guards and the hydra and merely being Overshadowed by Awesome - since only one person can wear Hermes's sneakers. Rule of Cautious Editing Judgement really applies here.
Which of the movies is better or worse and which is the most faithful adaptation.
There's something particularly aggrivating with Hades's Adaptational Villainy. While his morality was ambiguous and he was actually decent in comparison to Zeus in the books, he's given the Everybody Hates Hades treatment from the writers. It certainly doesn't help that the Underworld is designed to look like Hell (literally), thus implying that he's the Devil.
Persephone's strange attraction towards Grover. Most incarnations show her to be Happily Married to Hades, but the film very clearly states that she's unhappy with him.
Evil Is Sexy: Medusa. Of course, she is played by Uma Thurman.
Nightmare Fuel: The scene in Medusa's shop. While it isn't that creepy in the book (there the scene is a lot less ominous up until the Uncle Ferdinand bit), the movie added a panicking woman whose husband had already been turned to stone and she's hysterical with fright. She eventually gets turned into stone too, and the only mention that's made of it is when Annabeth explains how she scraped her wrist (the woman had been holding her wrist when she had been turned to stone and Annabeth had to pull her hand free).
Counterpart Comparison: Probably helped by them sharing an actor (George Salazar), but Grover is sometimes compared to Micheal from Be More Chill, due to being the main character's best friend, and having a heartbreaking solo song. ("The Tree On The Hill" for Grover, and "Micheal In The Bathroom" for Micheal)
One-Scene Wonder: Clarisse in "Put You In Your Place". While she later sings in "Bring On The Monsters", "Place" is the only song in which she's the main singer.
Win Back the Crowd: The musical has been much better received by the fandom then the movies
The Woobie: The main trio all get songs that present them as this: "Good Kid" for Percy, "The Tree On The Hill" for Grover, and "My Grand Plan" for Annabeth.
"The Campfire Song" highlights how the gods have ruined the lives of the campers in some way or another.