Harry Potter fandom has such an enormous fanfiction community, with so many common plotlines, that it's difficult to keep count.
A quintessential example, The Late To Hogwarts Fic: through some contrived effect, a young wizard doesn't get their invitation to Hogwarts at age 11, or is unable to go, until later age (usually when romantic plots would be more ok, so from fourth year up) — the other way is to introduce a transfer student (who's usually from the same country/general area and of the same background as the author, and is very similar to them in all aspects, only better). The latter type has the added benefit of 1) making up a new, totally awesome wizarding school 2) enabling the character to have a unique, totally cool wand 3) creating an explanation for unseen-in-Britain, totally super magic 4) allowing the character to have an inherited katana. Details not tying into the character's origins and entrance into Hogwarts usually follow the normal self-insert/wish-fulfillment procedure.
'Remus is cured of being a werewolf', or less commonly 'the Longbottoms are healed'. Often via a potion invented by Snape.
Harry/Draco stories invented the Male!Veela Fic. May be stretched to apply to other pairings. Within the Male!Veela Fic, it's not uncommon for veela!Draco to have a predetermined mate (and it's always called a "mate"). These seem to focus more on Hermione or a new character, either way, she'll put up token resistance before realizing how awesome and sexy Draco is.
Harry is sorted into Slytherin, Hufflepuff or Ravenclaw instead of Gryffindor. Slytherin is by far the most common. Expect Draco to be Harry's best friend while most of the Gryffindors, especially Ron, are dumb jocks at best and outright evil at worst. And his personality does a complete 180.
Hermione finds out that she's an adopted pure-blood and, of course, drops her adoptive family like a hot potato in favour of the new, improved version. Her new family will invariably be aristocratic and on the same social plane as the Malfoys, often resulting in her befriending Draco if not romancing him. Her newfound wealth, especially the clothing, will be described in loving detail. She will get a complete personality transplant, sometimes with the added feature of an Unnecessary Makeover, causing her to abandon her old friends and more critical reviewers to wonder why the author didn't just use an Original Character.
If she's shipped with Draco, there's a good chance it's the result of a marriage contract.
Occasionally she's revealed to be Draco's twin sister, in which case, she's usually paired with Blaise Zabini.
Along with her newfound pure-blood heritage often comes a set of magical powers surpassing anything in canon and sometimes flatly contradicting it, such as her being an Animagus able to transform into any kind of animal.
The gang, in whole or in part, time-travels to the time of the Marauders.
Sometimes, they go back to read the seven Harry Potter books to them (excluding Peter). Basically a MST being performed by the Marauders of the original Potter books as though they were fiction for the Marauders.
Eventually the "Reading The Books" fics became a kind of popular subgenre, no longer being limited to the Marauders, nor people from the Marauders time. Though many of them are MSTs, some have crossed them over into Time Travel or Redemption plots - though some of them do fall into the 10% worth reading. The most popular setting now is to have someone send Umbridge the books and fool her into thinking they'll give her leverage against Harry - she then reads the books to all of Hogwarts, though no more than a handful of them are liable to say anything over the course of the series. The Umbridge version has also been done with My Immortal (here, complete with Umbridge censoring the profanity) and Becoming Female (here).
Stories in which Hermione alone is sent back in time for some reason and invariably falls in love with Remus or Sirius or Snape used to be very popular circa 2005 to early 2007, but have since all but vanished. (It's not a bad concept, as long as that's not all that happens. Hell, Voldemort had been working on his power base for almost thirty years by the Marauder Era.)
A variant was also done in Child of Grace, except that the author skipped the sex and just had the memory Riddle be a nice guy all along, in no way responsible for the evil actions of his present self (ignoring things like by the time he made the journal he had already released the Basilisk).
In response to the medical finding that Purebloods have become so inbred that they have reduced fertility rates and/or are likely to give birth to Squibs, the Ministry of Magic establishes a Marriage Law which forces Muggleborns (and sometimes half-bloods) to marry pure-bloods. Often the law is actually a ploy enacted by Voldemort supporters/sympathisers still in power to punish or neutralise the heroes of the second Voldemort uprising. Used to end up pairing Hermione with Snape (since the concept originated as a fanfic-writing challenge on a Snape/Hermione mailing list), but sometimes includes other pairings. Since then, Draco has become the most common.
Despite Snape being a half-blood, prior to Half Blood Prince it wouldn't have been entirely unreasonable, given Snape's former Death Eater status, to assume he was a pureblood. The Marriage Law theme originated during that time period and just kept going after being Jossed.
If he isn't ignored entirely, addressing how Hermione already marries a pureblood has its own common explanations, either Ron insisting that she pump out endless babies just like his mother, Ron suddenly turning evil, or "Ron marrying Hermione to protect her from Draco's marriage contract would make the Malfoys REALLY MAD! Only Snape is strong enough to protect her!"... which doesn't really make any sense. Harry, being not technically Pureblood, is not an option. Sometimes worse options are brought up as possibilities.
Some Marriage Law fics state that partners will be chosen by the government. One fic even included a statute that the marriage must be proven consummated within a certain window of time.
This scenario often involves Hermione's extensive use of the Time Turner in book three making her half-a-year or more older, to make her old enough to qualify for being forced into marriage. (Forgetting/Ignoring that this is kind-of cancelled out by the time she spent petrified in Chamber of Secrets, and that she was only allowed to use it to attend classes that clashed with others (Divination and Muggle Studies), which adds up to two weeks at most.)
A 'Marriage Stone' declares two people as 'soul mates'. From there they must be married and live together in private quarters. Bonus points if they hate each other.
Harry has a twin, and it's he or she who is the real Boy/Girl-Who-Lived.
Alternately, the twin is hailed as the X-Who-Lived, while Harry is both the real Boy-Who-Lived and ignored.
Frequently in this scenario Harry runs away from home and Voldemort finds and adopts him almost immediately, treats him far better than his previous parents/caretakers, and is like the best evil dad ever.
A For Want of a Nail outcome, where Neville Longbottom is the Boy-Who-Lived, is explicitly mentioned by Dumbledore and Harry at one point hypothesizes what such a world would be like, practically screaming to be used.
Harry discovers he is the Lord and Heir of the Kingdom of Azkaban and that the Ministry owes him several centuries worth of backtaxes for using his prison.
After Voldemort wins (or worse, after a Pyrrhic Victory over Voldemort that leaves Wizarding England a depopulated wasteland), Harry goes back in time in some way to his school days so that he can use his foreknowledge to craft a better outcome. Occasionally a different character will do so.
A Very Potter Sequel has a kind of reversed version of this, with the bad guys trying this to undo the happy ending.
The "Dursleys abuse Harry" fanfic. In these fanfics, Harry's abusive childhood is usually treated more seriously than in the books. Expect rape to occur. Other common features of these fanfics:
This new level of abuse will not be retconned as having been happening all along. Instead, it will start when Harry is a grown teenager with wizard friends and a supposedly murderous godfather. Usually, Harry arrives back at Privet Drive for the summer and, for whatever reason, the Dursleys immediately decide to start beating/raping him. Often the catalyst is Moody's threats at the end of the fifth book, with Vernon developing a "they can't tell me what to do" attitude in response. Expect the story to begin In Medias Res, with a beaten Harry thinking back on how different the Dursleys have been this summer.
The Dangerverse is an exception to this. The abuse has been going since at least a few days after Harry ended up at the Dursleys, prompting Remus Lupin and Hermione's older sister to rescue him.
Vernon is entirely responsible for the abuse. Petunia and Dudley will be mentioned so little you'd think Vernon and Harry were the only people living in the house. If Petunia and Dudley do participate in the abuse, they'll essentially be Vernon's minions. Alternatively, Petunia has a Heel-Face Turn.
Harry is given an impossible list of chores, which he is beaten/raped for failing to complete.
Harry is once again forced to live in the cupboard under the stairs.
Vernon brutally kills Hedwig so Harry can't contact the Wizarding World for help.
Harry is rescued by someone, usually Snape. Thus, the abuse may be setup for a Hurt/Comfort Fic. Alternatively, it's setup for an evil!Dumbledore fanfic.
Despite being a grown teenager who has faced Voldemort several times, Harry can suddenly be bullied, abused, and intimidated as easily as if he were a helpless five-year-old. Harry never saves himself in these stories — he either dies tragically or gets to be someone else's damsel-in-distress. In stories where Harry does save himself from the Dursleys, he'll be a God-Mode Sue who beats the Dursleys into submission the second they lay a finger on him and therefore no abuse against Harry ever actually happens.
Hermione and Draco are Head Boy and Head Girl in their Seventh Year.
The "Head Boy and Girl's private quarters" plot device, where they have to share a suite of rooms, providing lots of forced interaction to fuel sexual tension. Used for Seventh Year Harry/Hermione and Ron/Hermione fics, Hermione/Draco Foe Yay, and even fics with a Mary Sue or Gary Stu paired with Harry, Ron, Hermione or Draco, and originated with the once-popular Percy/Penelope ship.
Most oddly, this is so codified that it seems to have been forgotten that the Head Boy and Girl are actually supposed to be top students. Instead, fanfiction makes it look like the selection of Head Boy and Girl is completely random. No one ever even wonders any more "How did Draco Malfoy get to be Head Boy?"
This device also turns up a lot in Lily/James fics as well as fics set in the Next Generation (usually with Rose/Scorpius).
The Yule Ball is restaged despite there being no Triwizard Tournament, sometimes with a Handwave and sometimes without. Hilarity and romantic hijinks ensue. Oh-so common before Order of the Phoenix — even The Draco Trilogy used it at one point. Apparently, it took the release of Phoenix for fanfic writers to finally grasp that the Yule Ball is not something that happens every year. Oddly, Potter Puppet Pals used this years after Phoenix when Snape writes in his diary about being at the Yule Ball with Lily Evans, although he was describing a dream he had had.
Harry and Ginny (or Harry and Hermione, or Harry, Hermione and Ginny, or Harry and Luna, or Harry/Anyone, or Harry/Everyone) discover they have a Soul Bond linking their hearts in eternal Twoo Wuv and their magical cores in eternal Bad Ass.
There's a school class which pairs the characters to learn how to be grown-ups, normally by making them live together as a married couple in the Room of Requirement, take care of a magically created baby, and (of course) fall in love. Because putting hormonally fuelled teenagers together, and making a child you plan to delete in a couple of months, isn't morally wrong. At all.
Turns out the Marauders weren't the real pranksters of Hogwarts. Oh no, it was Lily and her band of female friends, who were Just As Good As The Guys If Not Better. Lily joyfully out-pranking the Marauders would seem to contradict the fact that she canonically considered their pranks cruel, but that has not dissuaded anyone.
Lily's band of female friends appears a lot in the Marauder-era fics anyway, pranksters or not, at least in the Finnish Harry Potter fandom. Usually, the group includes a beautiful, loud, extremely popular girl and a more soft-spoken, very smart girl, who are paired to Sirius and Remus respectively.
Before Order of the Phoenix revealed more about Arabella Figg she was often portrayed as Lily's best friend and Sirius's love interest.
Harry's real father is Severus Snape, not James Potter (despite canon repeatedly saying that Harry looks almost exactly like his father did at the same age. Sometimes this is Handwaved, more often it's just ignored.)
Funny thing about this one. Like the Marriage Law fic, it was initially popularized by a challenge. These fics are often called Severitus fics, after the Severitus Challenge, which was issued in 2001 by, you guessed it, Severitus. The original constraints of the challenge called for Harry's appearance to gradually change to resemble Snape's over the course of the fic, with the idea being that his original appearance was the result of a potion/glamour/whatever. (They also called for Remus to come back to Hogwarts at some point.) The challenge was extremely popular, and the name stuck even after people started to drop some of the requirements. (Sometimes people call just plain old "Snape is Harry's father" fics "Sevitus" fics.)
The usual explanation is usually one of the following 1) Lily really loved Snape despite him being a muggle hating Death Eater whose ideology drove them apart, 2) James Potter never grew up and is an immature brat/jerk so Lily had a secret affair, 3) James Potter couldn't have children and Lily asked her friend to do them a favor. 4) Any number of inane ideas that completely ignore canon.
Ron the Death Eater, a situation so common it became the Trope Namer for the entire general phenomenon: Ron is so jealous and resentful of Harry that he spontaneously does a Face-Heel Turn. Often a result of Harry/Hermione shippers wanting Ron to Die for Our Ship. Alternately Ginny's been dosing the pair with Love Potion to steal Harry's money and fame. Or Hermione's the one doing it.
Most of the Tom/Ginny fics written by shippers who don't mind the extreme age difference between the two or that Ginny is barely 12 can be summed up as "Tom Riddle has sex with Ginny in the Chamber of Secrets before Harry shows up".
It's not so much the extreme age difference as the fact that Ginny is barely 12. (And the myriad other ways the 'relationship' would be wrong even if she was older.)
Harry and/or Hermione go to Australia to retrieve Hermione's memory-wiped parents and complications arise.
No one starts Hogwarts at the age of eleven anymore. Instead, original characters seem to stop by Hogwarts just for their final year so they'll be old enough for teen romances (and, presumably, so they'll be closer in age to the author). Sometimes there is a Handwave often along the lines of she already knows everything she needs to.
The same basic "Summer of Love after Fifth Year" theme is done quite frequently, with almost every thinkable female character from the series: Ginny, Tonks, Susan Bones, ...
And in the case of Tonks, she is usually assigned to be a guard near Harry's house (mainly as an excuse by the author to have her there) and quickly grows sympathetic to his guilt and sadness over Sirius's death. Within the two months before going back to begin his sixth year, they'll have completely fallen in love. The fact that she is six or seven years older than him at this point is often (understandably) brought up by everyone, that is if Harry and Tonks aren't seeing each other in secret.
Considering that Tonks ends up with Lupin in canon, who is something like 11 years older than her? The age gap really isn't a big issue. Second, Tonks is about as mature as the Trio pre-HBP.
Harry escapes Privet drive and Dumbledore's manipulations somewhere around 4th or 5th year to become Lord of the Vampires.
Harry actually didn't save Ginny when he destroyed Riddle's diary. She was actually still evil, but didn't let on. This can go two ways. In the unsympathetic route, this becomes Ginny The Death Eater. And then there's the sympathetic route, which amounts to "Ginny's evil, but that's all right because Evil Is Sexy and now she can be Tom's queen." (The latter type nearly always finds a way to revert Voldie back to his Riddle-era good looks.)
Harry discovers his parents' will(s) were never executed (usually because of the machinations of a Manipulative!Dumbledore, and often in the course of receiving his inheritance from Sirius post book five). By various means he corrects this, and becomes filthy rich and a Political Power. (Which is weird, because he's already filthy rich, and he doesn't do anything with it.)
Sometimes it will be discovered he gained legal adulthood and hence 'full access' to the vaults (despite him appearing to already have this in the books) and Potter and Black seats on the Wizengamot because he's in the Tri-Wizard Tournament, despite the fact that the Tri-Wiz has included plenty of underaged students before. It's just this time the school administrators rigged a external age line to keep them out.
These fics usually explain this- Harry's current vaults are a trust fund. (usually justified by noting that there is only money in Harry's vault, when other families have been known to store all sorts of thing in their vault
Dumbledore's a Manipulative Bastard setting Harry up as a sacrifice in order to defeat Voldemort (in more extreme cases with Dumbledore planning to take all the credit afterward). But Harry has managed to get emancipated and come into his own, with all the money and political power he needs to thwart both Dumbles and Riddle. In well-done instances, Harry must race to learn who to trust and what he needs to know in order to save himself before Dumbledore's or Voldemort's plans come to fruition; in poorly-written implementations, Harry effortlessly defangs and counters all opposition, leaving them blinking in stupefaction in his wake.
One of the sure signs that you're in one of these fics is for someone to mention that Molly asking Ginny what platform they need to get to for the boys to go to Hogwarts is a suspicious act; the reasoning being that Molly wouldn't need to ask as she's already sent five sons and is in the process of sending #6, so obviously she was planted there to lure Harry into her devious clutches. The fact that parents all over the world play the Question Game (wherein they ask their kids to answer questions the parent already knows the answer to, because the point isn't what the parent knows, but what the child knows) never seems to occur to authors of this type of story.
A few fics point out that where she was playing the question game is a violation of the statute of secrecy.
In addition to these, Harry sometimes gets a 'magical inheritance' that instantly grants him a Superpower Lottery of magical talents and abilities, with no effort involved. Expect him to be a Metamorphmagus like Tonks, an Animagus (almost always a Phoenix), and have several other skills and 'lost' forms of magic that nobody else in the series has, such as a Golden Patronus. In most cases, he has to go to Gringotts to receive it, which crosses over with the 'Harry-Is-Nice-To-Goblins' plot-line. Yes, he is informed of and given his true power and respected by Goblinkind just because he remembered Griphook's name.
Although, in the last book Griphook is indeed slightly impressed that Harry does remember him and his name. No magical inheritance or lost magic, though.
Sometimes Neville helps out in this scenario because, coming from a traditional magical family, he knows the secrets of the magical inheritance and because they were born so close together Harry and Neville will go through their "magical inheritance" at the same time.
Also very common in these fics is to have Ron, and often Hermione or Molly (but alwaysRon) revealed to only have been friends with Harry on Dumbledore's request. The idea of an eleven year old child pulling off this kind of long-term deception is laughable, and if Hermione is included it reaches the far side of the Gambit Roulette, given how her actual friendship with Harry began. Finally, it's not uncommon for Dumbledore to be paying Ron (and any others involved) with money embezzled from Harry's own Gringott's account, which would only increase the odds of Harry finding out the truth.
And all the above are often learned in the future, and then Harry does a Peggy Sue.
Sometimes, for cheap angst, after fifth year Ron and Hermione turn against Harry for no apparent reason.
A more recent theme is having a 'true prophecy', usually ending in a dark!harry or god!harry story.
Someone (usually Snape) is turned back into a baby or a small child. He then must be cared for by the other characters or, in some cases, one character in particular. All in an effort to make Snape cuddly.
In the course of the second task of the Triwizard Tournament, Gabrielle Delacour bonds to Harry and becomes for all magical and legal purposes his wife. Fortunately, due to the vagaries of Veela biology she is not actually the eight-year-old she appears to be, and shortly thereafter undergoes a perfectly normal delayed and accelerated puberty that leaves her physically at her real age of 14 or so. Can be found in both Harry/Gabrielle shipping stories and Harry/Harem stories.
Harry discovers he is bound by one or more marriage contracts signed before his birth or in his infancy, requiring him to marry a girl (or girls) he barely knows.
Harry's simple courtesies to the goblins of Gringott's mark him as uniquely different from the racist bastards that make up the rest of the Wizarding World, and they throw all their military and economic power behind him. This despite the fact that the goblins are pretty much always anti-social manipulative bastards in canon.
Matchmaker!Dumbledore. Wise, old Dumbledore understands how right the author's One True Pairing is and becomes a Shipper on Deck, dropping irrelevant concerns like killing Voldemort. After all, if Dumbledore says Draco and Ginny should be together, then obviously it must be true, right? Naturally, never combined with Manipulative!Dumbledore because manipulation which brings about the author's OTP is obviously good. Often leads to the "Head Boy and Girl's private quarters" plot.
During the Three Year Summer, it was common to see Arabella Figg turn up as the new Defense Against the Dark Arts teacher, typically with a portrayal as the most badass Cool Old Lady ever. After Order of the Phoenix established she was a Muggle Born of Mages, this plot died a quick death.
In a growing number of fics, these uber-wizards are the witches and warlocks of Bewitched.
The idea of a 'double witch' is mocked in The Methods of Rationality, as a wish fulfillment fantasy for young witches and wizards who want to have "special magical powers". Hermione realized at this point that witches and wizards don't really seem to appreciate their powers at all.
One of the characters (usually Harry or Draco or Hermione) become an animagus or are transfigured into an animal, and hide in plain sight as another character's pet (usually Harry or Draco or Hermione).
Often a character is turned into a snake so only a parselmouth can understand them.
The Dursleys dump Harry in the woods as a toddler and he's raised by snakes, grows up only knowing parseltongue, and becomes an accidental snake animagus. Usually a set up for manipulative!Dumbledore, dark!Harry, and for Voldemort to Pet The Snake.
The "what if Voldemort won?" fic. This leads to a Dark Fic set in the bleak Dystopia he has created, but all too often it's just an excuse to set up a kinky Lemon in which Hermione is forced to be Draco's Sex Slave or somesuch. Post-Deathly Hallows, the non-lemon versions tend to instead take place during that book when Voldemort did win for certain values of the term, but not permanently.
Diary fics where the diary writes back, like Tom Riddle's. Sometimes this is for a villain's own nefarious purposes, like it was in the one canon example, but sometimes they rely on the premise that these diaries are as normal in the wizarding world as portraits that talk.
Imagine the advertising for that. 'Hi there, parents. I'd like to put my memories in this book, to live under your daughter's pillow and have her tell me all her secrets, and I'd advise her what to do!'
A common one is to introduce a Mary Sue with an additional prophecy. This prophecy usually concerns whether Draco sides with Voldemort or Dumbledore, how only she can guide him in this choice, and how Harry's prophecy is entirely dependent on this one. Expect the prophecy to only use vague terms about "light" and "dark" and animal imagery (Draco is usually the "Dragon," while the Mary Sue is normally the "Phoenix"), despite the fact that the original prophecy didn't bother with obscuring imagery.
God!Potter is very common. The general outline usually has Harry returning for his next year at Hogwarts after being trained by Elves/Demons/Vampires/Goblins/Time Lords and already having powers far more advanced than anyone in the school. Staples include limitless shapeshifting, wandless/wordless magic, "almost Unforgivable" curses, fluency in a secret language and floating runes. They almost always follow the plot of a canon book, but Harry solves each problem with his new powers while hiding his real potential.
There's quite a few fics where the Hogwarts students take a mandatory trip to the U.S. Sometimes they're visiting a wizarding school in the States, and other times it's just a plain old Muggle school.
Everywhere Else Is Full: A common trope in both the original books and fan fiction. Whoever gets top dibs on Hogwarts Express compartments, it's not main characters. Inevitably, Everywhere Else Is Full, so they'll have to share a compartment with Plot Relevant Character. In the original series, this trope is used to introduce Ron in the first book, Lupin in the third book, and Luna in the fifth book. The exact words "everywhere else is full" are actually spoken by Ron in the first book/movie and by Hermione in the third movie (not the book version). In the world of fan fiction, it's pretty much reached the level of Share Phrase.
Harry buys a super-trunk that's Bigger on the Inside and has it's own apartment, kitchen, library, potion's lab, training room, and Quidditch field.
Harry uses potion-steroids to fix his appearance, give himself a growth spurt, fix his eyesight, and grow his hair long. And he surgically conceals or removes his scar. This renders him unrecognizable to Dumbledore and his (once) closest friends.
All things considered? The first 7 probably would have been better parents. The first 6 are at the very least non-abusive. As for wild animals, just try attacking the kid of any wild animal. When you get out of hospital, then say they would be worse.
The only person who the fandom agrees would be worse would be Umbridge.
The funny part for this kind of story is that, all things considered, the Dursleys are actually the best option for Harry growing up. Why? A charm cast over Harry that ensures protection against Voldemort provided he stays at the Dursleys until either he becomes 17 or leaves the house.
As pointed out in several fics, the so-called super special blood protection is actually pretty worthless considering it doesn't protect against Dementors, Death Eaters, or the Dursleys themselves and even the claim that it protects against Voldemort seems kind of dubious considering that:
1. Harry never considered Privet Drive to be his home, it was merely the residence he was tortured in, so there is some question whether the wards ever existed at all.
2. Harry never received any "love" at Privet Drive and these super special bloodwards were supposed to run on the "Power of Love".
3. Voldie takes Harry's blood for his resurrection rendering the "blood protection" useless.
4. The Reveal that Harry is Voldemort's horcrux and since the wards are keyed around Harry, old Voldie may have been on the Privet Drive "invited in" list from the very beginning.
5. If the wards exist they are only keyed to the house and Harry doesn't live his entire life inside the house—indeed he is often outside going to school, doing chores like washing the car, weeding the garden, mowing the lawn, cleaning the drains, and running to the store on errands for the Dursleys. He's also often outside, in the park trying to avoid the wrath and vitriol of Vernon and Petunia and trying to escape being beaten to a bloody pulp by Dudley and his gang.
Instead of trying to kill Harry on that fateful night Voldemort kidnaps him and raises him as his own son, training him to become a weapon against the light.
Adult!Harry is sent back in time to the 1940s to raise young Tom Riddle and give him a loving home so that he won't become Voldemort.
Lily Evans wasn't really a muggleborn but she was secretly a magical creature and/or a descendant of Slytherin/Grindelwald/The Hogwarts Founders/Morgan Le Fey/Atlanteans and so Harry has some sort of secret inheritance (magical or political) waiting for him by the time he reaches a certain age.
Harry potter is often Crossed over with Doctor Who, in which case two things will happen: the Doctor will be suprised by Hermiones' timeturner, and, when the Doctor is in his tenth incarnation, he will be mistaken for Barty Crouch Jr.
Voldemort/Tom Riddle ( usually in horcrux form, most commonly the diary or the scarcrux) secretly mentors Harry leading him to join the dark side and helps him to either get away from the Dursleys and/or put the filthy muggles in their place.
Sometimes this will be justified with a Retcon that it wasn't the Dark Arts that destroyed his appearance but Voldemort is a metamorphagus or partial-snake animagus and he's looking like that on purpose so that his followers will deify him as a more-than-human Lord of the Dark. And because it's kind of hard to rule by fear when everyone's drooling over you (or he may have done it in an attempt to dissuade Bellatrix.)
Lucius Malfoy being in the Board of Governors even after being kicked out of it at the end of Chamber of Secrets. Particularly bad examples show him in the Board even during sixth year, when in canon he was either in Azkaban or an escaped felon at the time.
Lucius Malfoy as a Wizengamot member, even after Order of the Phoenix established he was not.
Nagini is somehow able to smell that Harry is a horcrux and tells Voldemort.
Hedwig is a white/snow phoenix in disguise.
Harry reasons that since simple spells they learn in class like wingardium leviosa can be deadly he's more than justified in learning the Dark Arts and spells that eviscerate people.
Alternatively, Harry argues that no spell is truly dark, even the Unforgivables that require the caster enjoy killing/controlling/torturing their victim.
The "owl, cat, or toad" rule is more of a guideline since Ron kept a rat and Dean had a tarantula. Some maintain you can bring any animal to Hogwarts as long as it's your bonded familiar.
Harry is the reincarnation of Salazar Slytherin, with all the memory-access, skill-acquisition, and personality-adjustments commonly implied in Reincarnation Fanfic. Often used with Tall, Dark and Snarky Slytherin who wasn't really a racist prick. Existence of basilisk is justified or handwaved. Sometimes it's Harry alone, sometimes three other students are the other three founders. Expect Hermione to be Ravenclaw and Ron to be Gryffindor (unless it's a Ron the Death Eater fic.)
Often borrows pieces of the Founder's Heirs plot, where four people, usually including Harry, are direct descendants of the Founders and get special magical skills/talents/powers due to said heritage.
A time travel plot where Harry is sent back to The Founders Era and becomes Salazar Slytherin is quite common as well.
Salazar Slytherin actually left the basilisk as a weapon of last resort to protect the school from witch hunters or Vikings raids or something and it was only his descendants that twisted its purpose.
Gryffindor and Slytherin were much closer than history remembers but their relationship wasn't accepted by their peers and Slytherin chose to leave in order to preserve Godric's reputation seeing as he was already cast as the villain for being "Dark" and a Parselmouth.