The Kite Runner: Amir standing up to Assef before the latter can rape Hassan, instead of running away as he did in canon.
Carrie: These stories tend to involve some new kid who comes to school and is nice to Carrie, often befriending her or at least smacking around her bullies. Tend to have similar powers so they can share in her experiences. These fanfics also tend to have them either stop Carrie's rampage or actually be complicit, helping her kill the rest of the prom students.
The Dresden Files fandom is really into Gender Bender, particularly of Harry, in an attempt to offset- or at least explore- his sexist tendencies in the early days of the series.(Pairing Female!Harry with Marcone is optional but frequently done, as Harry is heterosexual in canon.)
Another massive trend is the Sex Kitten Winter Knight!Harry, taking place in some amorphous time after Harry's become the Winter Knight. They, similarly, offset or explore Harry's canon attitudes on sex (I.E., not having sex with a new partner while still trying to save an old one, not having sex with monsters trying to eat or manipulate you, and not having sex with minors), which parts of the fandom see as repressed or, at times misogynistic.
Percy Jackson and the Olympians: Half-blood goes to camp. Half-blood gets amazing powers. Half-blood saves world. Simmer for five minutes. Serves thousands.
Probably lifted straight from other fandoms, the demigods/Olympians read through any of the book series or watch the movie. Normally involves a contrived setup.
Annabeth goes to Goode High stories are a subgenre of another increasingly prevalent subgenre of High School AU. The latter of which goes through great pains to fiddle with the ages of everyone so it isn't as creepy when normally-10-year-old Nico is in love with 15-but-actually-22-or-so Thalia.
Crossovers with the Kane Chronciles are quite common as well
Another common plot is having Percy inexplicably kicked out of/leaving camp for any number of reasons, then joining Chaos for revenge. Nine times out of Ten, there will be a Mary Sue in there somewhere.
Several fics write the stories of demigod children whose godly parent isn't part of the Olympian gods. What would a child of Nemesis, goddess of revenge, be like? A child of Hypnos, god of sleep and dreams? A child of Hecate, goddess of magic? It helps that the sequel series, Heroes of Olympus does focus more on the other gods.
Writers for The Silmarillion, specifically Maedhros/Fingon shippers, are practically obligated to write hurt/comfort fic based on Fingon's rescue of Maedhros from Thangorodrim. In all fairness, it's pretty much the last moment in the book for them which isn't utterly tragic.
Stories set during the years when Maedhros was Lord of Himring before the Battle of Unnumbered Tears are also very common, usually involving a visit by Fingon on political business.
If you happen to be a fanfic writer for The Children of H˙rin, it's almost obligatory to compose a Beleg/Túrin hurt/comfort fic at some point in your career. Somewhat justified, though, with canon lines like this one:
Then at the egging of Androg they left Beleg tied to a tree without food or water... but he said no more to them. When two days and nights had passed in this way they had grown angry and fearful, and most were now ready to slay the Elf. As night drew down they were all gathered about him, and Ulrad brought a brand... but at that moment Túrin returned... and he saw the haggard face of Beleg in the light of the brand. Then he was stricken as with a shaft, and tears long unshed filled his eyes... "Beleg, Beleg!", he cried... at once he cut the bonds from his friend, and Beleg fell forward into his arms... At first he gave heed only to Beleg, and tended him with what skill he had.
Also, fics featuring Faramir or Legolas being abused by their respective fathers are ridiculously common.
Also highly popular (so popular that it's lampshaded in nearly every fic involving it) are the 'hunting trip gone wrong' fics. Usually they involve a younger Aragorn and Legolas (who are generally accepted to have known each other before the Council of Elrond), with occasionally Elrohir and Elladan added in, going on some hunting trip or another, being attacked by orcs/spiders/assassins/humans/wolves/anything that can do anybody harm, and being incapacitated enough to not be able to return to Rivendell in a timely fashion. Not that all of these fics are bad by any means, most are actually quite well written, but they do tend to be highly common.
Legomances were nearly ubiquitous in the early/mid 2000s, where Legolas was paired with an original female character (or sometimes an out-of-character Éowyn or Arwen). Arranged marriages were a common plot point, and the girl was sometimes - though not always - a Tenth Walker. She's almost always shown as being a good fighter regardless of her actual role in society, was always an elf, and often got into a Slap-Slap-Kiss style of relationship with Legolas before actually falling in love.
Similarly, Aragorn/Legolas stories that either vilified or completely ignored Arwen were incredibly popular and can still be found on fanfiction sites. Often this involved M Preg or dealing with the prejudices of their families.
Many, many The Hobbit fics find a way for Thorin, Fili, and Kili to surive the final battle. Those that don't often involve reunions in Valinor or reincarnation to bring the characters and their love interests back together.
A subset of these fics then has The Hobbit cast having to deal with the One Ring before or along with the LotR trilogy characters and events.
Fics with characters from another fandom put in Lyra's world of the His Dark Materials trilogy, not necessarily involving any characters from the trilogy, so that the author can demonstrate what shapes they think the characters' daemons would be, overlooking the fact that traveling to that dimension will not necessarily cause your daemon to manifest, but being born there will, so it only works as a whole-cloth Alternate Universe. Not the same as Common Crossover, because any fandom can be plugged into this scenario.
The fandom itself is full of Fix Fics about getting Lyra and Will back together again some years after their parting, often through some fourth magical item that the story is named after.
The Lion, the Witch, and the Wardrobe fics will generally either have a fifth member, always a girl, joining the original Pevensies or else meeting them once they arrive, where she aids or betrays them regarding the White Witch and Aslan, or else she meets Peter or Edmund during the Golden Age. She may be a fifth sibling, a cousin, a friend, or even a transplant from the 21st century, but it's always a girl.
There is also more than enough Susan/Caspian and Lucy/Caspian Shipping to fill two fleets.
Another common one is for Keladry of Mindelan (who became the first female knight after Alanna) to be kicked out after her probation year as a page and follows her life from there.
The Hunger Games has an awful lot of 'Random OC who may or may not be a Mary Sue ends up getting reaped'. Some better writers will play with this by not having the character actually win, but as there are only two possible plot permutations the entire thing ends up horrifically overused.
And of course there's plenty of Alternate Universe Fics that ask "What if Rue/Thresh/Cato/Clove/Foxface/Marvel/Glimmer/Boy From District 7/etc. had won?" and explore the rest of the trilogy from that character's point of view (or take the events post 74th Hunger Games in a completely different direction).
There's also the "Submit-Your-Own-Tribute" story, in which an author rounds up 24 characters submitted by other THG fans and writes them into the Games.
A lot of Darkest Powers fanfiction features werewolf Derek and necromancer Chloe realizing that they are mates. In canon, the only mention of them being mates was made by Liam, who was taunting them. Also, in the main series, the werewolves do have mates... but it's nothing as huge, compelling, or life-changing as it's often made out to be in DP fanfics. Rather, the only time it's treated as a somewhat big thing is in Personal Demon, when Karl attempts to explain to Hope that the reason he left her is because he was terrified by how completely attracted to her he was (and still is).
Writing about the kits of one of the main characters is an approach used often, as this happens in Warriors canon.
A kit is hated and neglected by its Clan for no reason (or a clearly shallow, arbitrary reason) but Shows Them All his purity of heart/badassery/worthiness and rises to become the greatest leader ever (Prophecies about the kit are optional but common). Both before and after a canon book used this template, it wasextremelypopular.
A common plot in this fandom is "The Socials jump a random female OC, leaving her beaten and unconscious in the vacant lot. One of the greasers finds the girl, takes her to the Curtis household to heal her, falls in love with her, drama ensues."
The gender-flipped version of the previous plot is also prevalent, "One of the greasers is badly beaten after a rumble and a random female OC finds them and nurses them back to health."
You also have fics where a girl helps give Darry a life outside of work/Help Sodapop get over Sandy/Help Ponyboy find love after Cherry Valance/becoming a nurturing mother figure for Johnny/"tame" Dally and lead him to the path of good citizenship/Keep Two-Bit in a steady relationship/Make Steve feel like a better person inside.
Fics that go "What if the Curtises had a sister? How would that have affected the book?"
Girls from the future getting transported into the past (May or may not involve getting sucked into the book/the movie) or the boys getting transported in the future.
A Soc girl who is nicer, prettier, and more understanding to greasers than Cherry, who thinks the feud between socs and greasers is pointless, and single-handedly brings down the segregation between the two classes.
The fandom is extremely fond of modern AU fics, generally focused on Les Amis as some sort of political activist group, and often turning the ridiculous amounts of Ho Yay in the source material into canon relationships (Enjolras/Grantaire being by far the most popular, followed by Jehan/Courfeyrac).
Post-Barricade fics dealing with the survival of one or more of les Amis (besides Marius) are common in non-modern fics as well. Most fics for the fandom also focus on les Amis to the exclusion of the older characters, and stories involving Fantine, Valjean, or Javert that don't involve the latter two in a relationship are extremely rare.
Fics that spare Eponine are also very common, and typically take two forms: either she's paired with Marius, sometimes but not always with a corresponding bitchification for Cosette, or she platonically moves in with Marius and Cosette and is paired with one of Les Amis who also survived. These fics also have a tendency to kill off Eponine's much less important sister Azelma instead of her.
There is a lot of (usually Valjean/Javert) fic where Javert survives his suicide attempt (often because Valjean prevents it or pulls him out of the river) and then has to figure out how to live with his change of heart/worldview instead.
Modern AU writers are fond of making Cosette and Enjolras into siblings. It's especially popular in Valjean/Javert fics, probably because the idea of Javert acting as a father figure to Enjolras is hilarious.
Many A Song of Ice and Fire fanfics have Cersei give birth to one of Robert's children. Said child ends up becoming all the good that is missing from Joffrey - and either fights his brother for the throne (which Robert bequeaths to him instead of Joffrey) or ends up inheriting the crown when Joffrey dies.
Also popular are fics where Sandor and Sansa aren't separated during the Battle of Blackwater. Either he choses to stay or Sansa flees from King's Landing with him.
AUs where Robb married either Margaery Tyrell, Roslin Frey, or an OC instead of Jeyne Westerling.
There are many fics detailing Robert's Rebellion, the events leading up to it, and/or the immediate aftermath.
There are quite a lot of Haruhi Suzumiya fics out there in which the Data Integration Thought Entity decides to punish Yuki for gaining emotions by deleting her — never mind that the Entity would gain nothing from this and it knows exactly what the consequences would be for itself if it tried — in order for Kyon to go berserk, tell Haruhi that he is John Smith and the other Brigade members are aliens, time travelers and espers, so that Haruhi can become aware of her Reality Warper powers so that the two of them can go on a Roaring Rampage of Revenge in Showy Invincible Hero fashion, rescuing Yuki along the way. If the fanfic writer happens to ship Kyon/Yuki, then it is practically obligatory for them to use this plot at least once, as it is a popular way to get Haruhi out of the picture without resorting to Die for Our Ship.
After Disappearance, it was not all that uncommon to see some fanfics about what life had been like had Kyon stayed in the world Yuki had made to give everyone a normal life. This will almost always involve Kyon/Yuki in some way, shape, or form. The Vanishing of Nagato Yuki-chan is basically an official version of this kind of plot.
There are a number of Looking for Alaska fics in which, as Miles theorizes in-universe, the title character's death turns out to be her latest and most intricate prank.
Bridge to Terabithia fandom has an entire fandom-specific genre, LDD, short for "Leslie Doesn't Die." While over half the fics (including the vast majority of multi-chapter stories) in this relatively small fandom are of this genre, the exact reason Leslie doesn't die varies from simply being rescued to faking her death to her body never being found (easy to do, as neither Jess nor the reader/viewer ever sees it) to Terabithia being Real After All and being used to restore her by magic. It's also an unwritten rule that Leslie would grow up mindblowingly gorgeous, probably inspired by AnnaSophia Robb, who played her in the Film of the Book, doing just that.
After the canon story ends, Taylor tries to settle down and live a normal life, only for epic adventures to drag her back into vigilantism. These stories usually involve Taylor reclaiming her old powers or somehow getting new ones.