Before Season 2 stories about the mane six having to face the "Elements of Discord/Disharmony/Chaos", Evil Counterparts to their Elements of Harmony, were quite common.
After Season 2 and the reveal of Discord, spirit of chaos and disharmony, stories abounded with Discord escaping from his can again, either to continue to wreak havoc... or somehow falling into Defeat Means Friendship, despite him being one of the few purely evil characters in the entire series. The second of these became Hilarious in Hindsight given the premise of the Season 3 episode, "Keep Calm and Flutter On".
Alternatively, an even worse evil shows up for the mane six to thwart, most commonly yet another Sealed Evil in a Can that Celestia and Luna dealt with millennia ago.
Cross Over fics dealing with other characters getting transported to Equestria, almost always getting inexplicably "ponified" in the process. Doctor Who is the most common victim of this, thanks to a background character who looks like a pony version of the Tenth Doctor.
Whatever the case, Lyra Heartstrings usually gets involved, at least as a cameo, or in a more or less prominent role.
Said inverse was eventually made (partly-)canon with the Equestria Girls movie, wherein Twilight Sparkle gets transported to the human world.
First Contact type "Human in Equestria" stories are also fairly popular. Usually, the humans are space travelers, but Megan has frequent appearances as well.
The standard plot for getting the ponies themselves into anotheruniverse is famously "Twilight Sparkle tests out a new teleport spell, and it goes wrong". It's gotten to where Equestria Dailyexplicitly points it out when a fic doesn't do this.
It's not just "teleport turns into dimensional travel"; when Twilight tries out any new spell in fanfic, things will go south. One fanfic involving Twilight visiting a darker parallel universe had Equestria Daily note that the universe's existence isn't her fault: "Is that even allowed?"
It's popular to pit the gang from Friendship is Magic against updated versions of G1 villains, such as Tirac from "Rescue at Midnight Castle", and the Smooze from the movie.
Or, before the events of the story, Celestia turning evil and being sealed into the sun.
Corollary: Luna (or even Nightmare Moon) taking over where Celestia was de-throned.
Rainbow Dash having an accident and, irreparably or not, injuring her wings, requiring another character to care for her (may or may not involve shipping). At one point these were so ubiquitous that Equestria Daily outright banned them. Then there was the actual episode where Rainbow broke her wing and was confined to a bed... (However unlike all the fanfics, her wing healed in only a few days, and she spent her time in a hospital bed, not a friend's house.)
Fluttershy nursing a hurt character back to health (which has some basis, given that she's the Element of Kindness), often overlapping with the above. Even Trixie is in one of these.
"Character X becomes an alicorn" has always been a popular plot, as it allowed exploration of what it meant to be an alicorn, having to deal with power and immortality, and what the Canterlot royalty was like. While all of the mane six have undergone it at some point, Twilight was by far the most common. And then it actually happened.
A popular theme in these has Twilight Sparkle (and possibly the other members of the mane six as well) turning into alicorns and taking over the rule of Equestria from Celestia and Luna, who have grown tired of immortality.
It used to be common for Derpy to be The Woobie, having a speaking disorder that typically made her speak gibberish and being The Klutz, which caused her to be ostracized by other ponies. This plot has since faded out, with more fans preferring Derpy as just being a klutz with a lazy eye instead of disabled, and the only remainder of it is Derpy calling Dinky "muffin".
In Derpy fics where she's the mother of Dinky, it'll often center around the hardships of being a single mother, and Dinky would either be embarrassed of her mom but learn to accept her by the end, or would be the helpful daughter.
Many a Bonbon and Lyra fic are about Lyra being unable to get a job, as she's a musician in Fanon.
"Luna reintegrates into Equestrian society" stories became endemic in the fandom almost since day one, even after the actual episode which featured it, with Progress only being the most well known. Almost all contain some form of The Atoner mixed with Adorkable, No Social Skills and Fish Out of Temporal Water, and most draw some sort of parallel to Twilight's situation, if not outright having Twilight help her adjust. "Luna Eclipsed" confirmed all of these and further strengthened their prevalence, although later ones drop the fan-made Shrinking Violet portrayal and adopt the official Large Ham portrayal.
Any fanfic that is centered on Scootaloo is usually going to make her an object of pity; either it will be about her inability to fly or her lack of any known family. She often ends up being comforted or adopted by Rainbow Dash as a result. The last part [[has become canon, to an extent.
There's a surprisingly large number of fanfics about the background ponies Vinyl Scratch and Octavia—who, being a modern techno DJ and classical musician respectively—somehow have to work together, whether they go to the same university, get assigned the same radio show, or become roommates. Hilarity Ensues. Shipping also occasionally ensues.
Alternate Universes where the Elements of Harmony are assigned to different bearers. This may entail switching them around between the Mane Six, with different aspects of their personality coming to the fore (or else having different personalities entirely), or having other ponies answer the call in this 'verse. Partial replacements or combinations of the above are also possible. Yet another Fandom Specific Plot that became canon (although it was a result of a magical mishap which nearly destroyed the character's livelihoods).
"Character X was a changeling all along" fics exploded in popularity after the changelings were introduced, though their ubiquity made them something of Dead Horse Trope before long.
While we're talking about Changelings: there's a good number of fics revolving around a Changeling turning up in Ponyville sometime after the Royal Wedding...and one can safely bet that Fluttershy will be the first to find said Changeling.
One fiction parodies this concept in general (though it was written as a parody of one specific fic), by making everypony a changeling (except Celestia). A three-way changeling civil war ensues.
Speaking of ponies being Changelings, there are also a number of fics that seek to exploit the sheer Paranoia Fuel that Changelings represent, either by having them attempt to reinvade, or retconning the original invasion. But instead of the Zerg Rush used by the Changelings in the actual show, they make full use of their shapeshifter abilities to selectively replace ponies (including major characters), planting deep-cover agents in Equestrian society and bring about its downfall through the use of full-on infiltration.
Trixie becoming a laughingstock and forced into destitution following the events of "Boast Busters" and seeking revenge on Twilight. Bonus points that now this is canon with the episode "Magic Duel", though not so much the severity of the first two.
Trixie being considered for Luna's apprentice. This in turn produced fics in which Trixie became the leader of a rival mane six.
A common subject is the possibility of a Griffon kingdom/empire that rivals Equestria in power, and what the relations between ponies and griffons are like.
"Spike gets turned into a pony" (often a pegasus, for some reason) is also pretty common, either accidentally or deliberately in order to impress Rarity. Presumably because 1: It makes Shipping easier and 2: It's a good setup for a Be Yourself Aesop that wouldn't be out of place in the actual show.
Having crossover fanfiction wouldn't usually be considered a Fandom Specific Plot, but the MLP fandom has a disproportionate amount of them compared to most for some reason. It may have to do with one of the fandom's most popular fanfictions also being a crossover.
It is very common for Equestria to serve as a sort of afterlife, with a character finding themselves in Equestria after they died in their original story.
In particular, Portal and Team Fortress 2 crossovers are extremely common, largely because Valve's games really helped spread the fandom in the early days.
Or Wheatley and the Space Sphere meet Luna out in space.
So are Doctor Who crossovers, simply by the deal of having the TARDIS land in Equestria (which, granted, is also a Fandom Specific Plot amongst DW fanfics for crossovers). Also, the Doctor almost always becomes his pony self, if he isn't already, and either Twilight or Derpy are his companions.
Twilight Sparkle grows old. This is a common topic as both her mentor and her little brother/assistant can expect to outlive her by centuries.
It isn't uncommon to have stories of the other members of the mane cast getting together in their middle/old age and discuss just how much things have changed over the years and what sorts of issues they are facing now that they're older and not as active as in their youth. How the characters achieved their dreams, what their descendants are doing, who got paired up with who and the deaths of loved ones are common topics.
Other stories that take place in the future may also be about the adult lives of characters who are children or babies in the actual series.
Ever since "The Cutie Mark Chronicles" and "Party of One", Pinkie's straight-haired mode has been something of a fixture of the fandom, with the change of Pinkie's mane and tail from fuzzy to straight indicating some form of emotional trauma. Equally frequently is the tendency of writers to refer to her straight-haired form as "Pinkamena" to distinguish it from the puffy-maned Pinkie.
Diamond Tiara is a Closet Geek. Or she's at least hiding something that would ruin her reputation. Another common trend involving her would be having to move in with the Apple family and having to deal with Applebloom.
