The Land Before Time fanfiction. Considering that the series is set in a relatively limited environment (prehistoric times), whenever one writer does something different that gets respect, you can expect to see a lot of clones pop up. Such types include war stories, time travel stories, crossovers. Not that these fics are necessarily bad, but they tend to all come at once as trends, resulting in you finding that all new fanfics lately are centred around Chomper perhaps, while you were particularly hoping for something based more on the main five characters.
Count the number of Pokémon fanfics which focus on an original trainer, who sleeps in on the morning when they're supposed to receive their first Pokémon, arrive at the lab only to find that all the usual starters are gone and instead get a completely different starter. Very often a rare and valuable Pokémon. Almost always an Eevee. Their goal is always to become a Pokémon master. To be fair, this is the purpose of the main series. The use of Eevee has generally slowed down, but OC journey fics are still the same even without Eevee.
There are sometimes fics that appear to be OC journeys at the start, then have a twist at the end that makes you think "what the heck..." until you re-read it. Then there are Pokémorph fics where a human turns into a Pokémon with no prior warning...
At one time, there were far too many Cardcaptor Sakura fanfics that blatantly copied each other; as soon as someone comes up with something unique and interesting, a wave of people start ripping it off right down to the last joke and plot point.
According to the Reunions Are A Bitch universe the Colonials of Battlestar Galactica were huge wankers before the Cylons kicked their collective arses to the curve, what with the whole starting a holy war against Earth thing. Naturally, the story was so popular that now there are dozens of different fics detailing the Colonials getting in over their heads while trying to Take Over the World. Most commonly these involve crossovers with Halo or Stargate SG-1, but other types do exist.
Cassandra Claire's Very Secret Diaries for The Lord of the Rings fandom inspired several copycats in several fandoms, some acknowledging their inspiration coming from her. Even to this day, several years after their creation you still see some of her more infamous quotes scattered about fandom.
Speaking of Cassandra Claire, The Draco Trilogy started (and named) the Draco in Leather Pants portrayal of Draco Malfoy. It also started or at least popularized some other Harry Potter fanfic clichés, like making Draco more sympathetic by portraying his father as an abusive bastard. The funny thing is, the vast majority of young fanfic writers who currently use these clichés have likely never heard of Cassandra Claire. They just copied them from earlier fanfics, who copied them from still earlier fanfics and so on, going all the way back to The Draco Trilogy. Cassandra Claire also popularized the idea of certain characters being "Magids", which are especially powerful wizards. When The Draco Trilogy was at its height, the concept of Magids was borrowed by numerous other Harry Potter fanfics, but now it's a Forgotten Trope.
75% of all Dragonball Z fanfiction starts with an alien spaceship crashing down on Earth, and ends with a character using a new form of Super Saiyan to beat them. It will also be set during the 3 months before the Androids, the 7 years before Cell or the 10 years after Buu. Other common elements include Vegeta or Broly getting possessed by Babidi, Buu or Cell absorbing someone new or each other, Saiyans inexplicitly growing back their tails, Bardock being rewritten as an honourable anti-hero and Future Trunks travelling to New Namek to use the Dragon Balls there.
The current trend among people who write Troll Fics (or who just want a lot of reviews and attention and don't care about the quality of their work) seems to be to copy My Immortal's format, issues with spelling, My Chemical Romance references, and Goth Sue main character.
In 2001, Potato Salad of Fanfiction.net wrote Ask Sonic (and other associated characters), which had reviewers sending questions through reviews or e-mail and having Sonic the Hedgehog and the rest of the cast answering them as fan mail. It became very popular and soon numerous Ask Fics began to show up on Fanfiction.net until it and other interactive fanfiction were banned from the site... not that it stopped anyone.
Nearly every Harry Potter story in which Harry is betrayed and sentenced to Azkaban follows the exact same template: Most of his friends repeatedly Kick the Dog, and instead of dying or going insane, Harry becomes a God-Mode Sue in prison and later wipes out Voldemort and all his Death Eaters with a single attack. Depending on the author, he then proceeds to kill everyone else as well or spend the rest of the story brooding with those who stuck by him (usually Snape, Neville, Luna, and a few others), while those who betrayed him plead for his forgiveness.
A relatively minor example: while Child of the Storm, a crossover between Harry Potter and Marvel Cinematic Universe, didn't even come close to inventing the idea of one of Harry's being someone other than Lily or James, or even Lily or James being a Composite Character with another character (in this case, James was Thor), but its success did spawn a surge of imitators, a number of which copied its Mega Crossover attributes too - certainly, quite a few are decidedly up-front about what they're following on from.
Eiga Sentai Scanranger was created to let its author complain about Power Rangers, yet seems to have taken quite a few cues from the imported show. Interesting is one scene where the blue ranger flies off the handle at the changes made in Power Rangers Dino Thunder, but no mention is made of how the character concepts for Mesogog and Elsa are almost identical to two villains from Scanranger. The Big Bad is a former colleague of the team's mentor sharing his body with an evil presence, and the lone female villain wears black leather and spies on the heroes via her job at their school (Scanranger did it first).
NGE: Nobody Dies have spawned a whoppingly large Expanded Universe around it. The author of the original even made some of these canon, like Ichi's Birthday Party, The Story of NERV-Alaska and The Kei Files.
When Total Drama Alphabet appeared, with its conceit of one contestant name for every letter of the alphabet, a host of imitators followed.
The Mass Effect section of fanfiction.net was known for having relatively few self-inserts. Enter Herr Wozzeck with his self-insert Mass Vexations. After that, a great number of self-inserts began to show up around the website, with varying results.
Once upon a time, a fanfic writer called Roxius wrote a Crack Fic for Code Geass titled You Got LuluRolled. This spawned many similar Crack Fics in several different fandoms, not all of them titled with some variation of "You Got [Character Name] Rolled!" Most of the imitations are actually derived from superstarultra's Haruhi version, You Got HaruhiRolled!, which is generally considered superior to its inspiration.
Quite a few The Familiar of Zero fanfics have been made with the premise of 'What if X got summoned instead of Saito?' All of them were complete crap, at least until gabrielblessing's The Hill of Swords came along with its lauded execution of the concept. One well-timed Plot Bunny thread later, SpaceBattles.com's Creative Writing forum was flooded with copycats. A fair few of these, such as Unfamiliar, Zero's Shock and The Shadow of Zero match or exceed the original in quality.
Since the fanfic Progress became popular, it has become commonplace for other writers to portray Princess Luna with a Companion Cube abacus.
Same with Cupcakes. Ever since it became popular, there have been many other My Little Pony: Friendship Is Magicdark fics set out to copy it's success.
The same also goes for My Little Dashie, which has become a Fandom-Specific Plot to the point where it's almost a Fandom Specific Trope. Nowadays, knock-offs of it with another pony in the place of Rainbow Dash are a dime-a-dozen, and not one of them has gained as much fame / infamy. It seems to have tapered off lately, partly because the story has developed a sizeable hatedom (which it doesn't really deserve) and partly because there's really only so much you can do with the plot and it's been worn absolutely smooth.
Often there are (usually unauthorized) fan-sequels written for popular stories, many of which have a completely different writing style and a tone that is sometimes quite changed from the original. The good ones stand out on their own so that they can be seen as an extension (or in some cases, a Deconstruction) of the original author's work, and have plenty of fresh ideas to make them just as likable if not more so. The bad ones are nothing more than cheap attempts to try and ride the coat-tails of another author, and are quickly derided as such by reviewers. The latter case is seen somewhat often by the myriad number of terrible Past Sins and My Little Dashie sequels out there that hope to garner positive votes and reviews solely because they are set in the same universe as the originals.
Brony in Equestria stories (where the main character is usually male, has knowledge of the series and more often than not gets turned into a pony and has incredible powers) are so widespread and often of terrible quality to the point that Equestria Daily has banned them completely, and in general they are also frowned upon elsewhere unless they are very good. Go to the My Little Pony section of FanFiction.Net or Fimfiction.Net and you are almost guaranteed to find at least one or two stories of this type on the main page, due to the massiveLonging for Fictionland this series has. This is not to be confused with Human in Equestria stories, where the character(s) have no knowledge of Equestria or its inhabitants whatsoever.
An interesting trend has popped up amid the Mahou Sensei Negima! fanfic authors. After Shadow Crystal Mage popularized the idea of crossing over Negima with Lyrical Nanoha, a few other authors have taken up the mantle. Currently, Mage and two others are writing stories in which Negi is sent to Uminari as a teacher for Nanoha after the events of A's. Ironically, one of the copycats, Zephyr-Fiction's Equal And Opposite Attraction, has garnered the most press. Mainly this is because Mage has many more projects on his plate.
It helps that both Anemoi(Zephyr-Fiction) and Shadow Crystal Mage have a tendancy to bounce ideas off one another on this very site.
Everybody wants to write about what Sherlock Holmes did during his Great Hiatus (the three-year period during which he faked his death), right? It's been the subject of numerous pastiches and online fics. Aragonite a.k.a Marcia Wilson has made the argument that none of these stories, however, deal with Dr. Watson, Inspector Lestrade, and Scotland Yard, and the adventuresthey must have been having in that time period. So she wrote her own series, which she plans to convert into self-published books. Thus far, only one other Fangirl has taken up the torch, so to speak, but that's not the only thing Aragonite started. She was arguably the first Fan Fic author to give any serious depth to the men of Scotland Yard and use them as the protagonists, rather than using Holmes or Watson. She also bestowed the name Geoffrey upon Inspector G. Lestrade, and that name has gained popularity among both the canon and Sherlock branches of the fandom.
The Naruto fandom has started a following of Self Insert via Reincarnation fanfics - the Author Avatar/Original Character is reborn into the Narutoverse. Although the first noteworthy story to try this might well be Only a Moron by swabloo (which actually veers into Dark Fic territory at times), this trope is applied to Dreaming of Sunshine by Silver Queen. In addition to having a likeable, relatable person as protagonist (Shikako Nara is noMary Sue), DoS touched on other (innovative at the time, at least) elements: the person being "chakra sensitive", since having memories of their past life means chakra is an unfamiliar feeling, like an extra limb; trying to not change things too much because they're aware of how the ripple affect could render their knowledge useless; feelings of alienation from their friends (even while befriending even Naruto); not remembering all significant details due to fading memories over time; and most of all managing to focus on the protagonist without making it feel like the world (or story) revolves around them.
The nice thing is that many of these authors try hard to not copy the same formula, even with this sort of inspiration. For example:
In Little Acorn the character is blind, does not remember her past life beyond flashes, and instead of being the sister of a noteworthy or popular character, her older brother is Yoroi Akado (a Mauve Shirt at best in canon).
In Distorted Reality, the main character is given bumps at the start: she's a prideful teenager reborn into the Hyuga clan (more specifically, the branch family), and must deal with the Curse Mark, all that's associated with it, and the harsh life of a ninja. On top of that, while unike LA she remembers her past life, she knows nothing of the Narutoverse in return.
If you go onto the official fansite for The Thing, you'll find a lot of fanfiction of various sorts. Given that both films (but especially the 1982 movie) end on a very ambiguous note, a lot of fanfics will attempt to explain what happens next. The majority of these stories tend to keep to the very bleak ending of the movies, often involving one of the two survivors being revealed to be a Thing and/or a later encounter with the Thing. However, the story Unfinished Business was a deliberate attempt to break from this trend with a much more optimistic tone.
Voltalia has admitted to liking the idea of giving her favorite shippings mixed-gender twins, so much so that when she created mixed-gender twins for the popular Superjail!Foe Yay shipping Warden/Mistress, another author put them in upon the suggestion of a reviewer on FanFiction.net when she decided to write in the Mistress being pregnant in Extended Stay. It looks like they're going to be quite popular.
Rhyme and Reason, the very first Chip 'n Dale Rescue RangersFan Fic (written from 1993 to 1995 and published in 1996), first averted this trope and then played it straight. The aversion was that a story that long and well-written did not inspire other fans to write their own fics at first, but instead intimidated them because they knew they could never even achieve half of it. It was another story published in 1997 which finally kicked off the general Fan Fic writing, but Rhyme and Reason set the general tone for most CDRR fics to come — Darker and Edgier, giving the Rangers more serious cases to solve, more dangerous opponents and more emotional turmoil.
What About Witch Queen? has underplayed example, as Kai mentions that many young noble ladies in Arendelle started to dress like Elsa (which is more than a bit revealing, especially by their standards), to their families' chargin.
Fix Fics based on rather badly received episodes, such as "No Such Luck" and "Brawl in the Family", are also highly likely to spawn a horde of imitators.
Common Itachi/Sakura plots in Naruto fic involve Sakura running off to get Sasuke back from Orochimaru's clutches, and getting captured/used as bait for Naruto by Itachi. Many cite Metamorphosis as inspiration or influence.
Doctor Who fanfics have many fanfics about fans of the show ending up in the universe, one way or another. Reality, on Fanfiction Dot Net is currently one of the most popular ones involing a fan ending up right near the end of 'Rose', and expands on several characters and the episodes themselves. And arguably, Lost In Time, which appeared around the same time, which detailed a fan of the show that encounters Ten near the final specials before his regeneration. There is still a wave of Realistic Fic after several years, but not as well known.