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 before they started banning such stories.
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?"
Prior to his reformation, Discord was a prime choice for an Ultimate Evil-type entity when one was needed to act as the Big Bad or Bigger Bad of a story, due to his nature as a god of chaos and disharmony and even now being one of the most powerful characters in the series short of Celestia and Luna.
A combination of the below and above, was an Enemy Mine type where Discord would be forced to join with the Mane Six. Bonus points if it turned out the thing created Discord, or at least Discord knew what it was.
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. Though considering this is how Seasons 1, 2, 3, and 4 began, it might just qualify as standard series narrative at this point
Lampshaded by Pinkie Pie in at least one fanfic, asking after yet another threat from a thousand years ago reared its head, "Was that just a really bad year for you?" Celestia noted that the actual reason for this is rounding.
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 known as Doctor Whooves.
It's also Generally accepted that if G1 is the past of FIM in a fanfic, 9/10 times, Megan has a Religion in Equestria dedicated to her.
It's also become fairly popular to depict, or at least reference, previous generations of My Little Pony as actual historical events in Equestria. Sometimes this even leads to crossovers due to time-travel or other such plot points. G1 is the most popular, but there is some G3 equivalent material out there.
This particular Fandom Specific Plot is likely to receive a huge boost, since the ending of season 4 involves a confrontation with an actual G4 rendition of Tirek, the very first G1 villain, and also namedrops Scorpan, the original Tirak's lieutenantnote rendered here as his brother and equal, who pulled a Heel-Face Turn and allied with the ponies to imprison Tirek, then returned to their kingdom.
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... for three days, in a hospital.
Trixie is a common target for Hurt Comfort Fics, and multiple fics exist for her being taken in by (and often shipped with) every single member of the mane cast, but especially Twilight.
Unless it's a story where Everyone Is Gay, Rainbow Dash is the go-to choice for a lesbian or bi-curious character. Some fics in turn play with this by having her defy this stereotype and be offended by other characters making this presumption of her.
"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 (and the show itself) 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. LyraBon itself is a very common ship, and the pair is frequently presented as together or married even in random stories that aren't even about them.
"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 centered on Scootaloo is usually gonna make her an object of pity; it'll either 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. Her inability to fly has become canon, to an extent, as has Rainbow Dash taking her under her wing and acting as a big sister of sorts.
Due to the lack of visible family, "Scootaloo is discovered to be orphaned/abused/etc. and gets adopted" is practically a genre unto itself by now. Rainbow Dash is popular of course, but there are adoption stories with each of the main six, pairings among the main six, Ms. Cheerilee (it helps that in G3, Scootaloo is Cheerilee's little sister and their personalities don't change overly much for FIM) Lyra and Bon Bon, Original Characters, fellow classmates of hers who are aged up for the story, Celestia, Luna, and even Discord.
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 frequently 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 characters' 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 and spies.
Related to the above, some fics portray Shining Armor as a Shell-Shocked Veteran haunted by how he almost lost his wife and married a Changeling. Considering there's no telling just how long Chrysalis impersonated Cadance before she was discovered, some authors go for the most extreme interpretation of the Fridge Horror this invites.
Trixie becoming a laughingstock and forced into destitution following the events of "Boast Busters" and seeking revenge on Twilight. Bonus points since this plot became canon in the episode "Magic Duel".
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's 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.
Also, because of the ease with which a crossover plot can be written this way, many, many crossovers involve some spacefaring civilization making planetfall on Equestria, leading to a First Contact type of situation.
Now that Twilight is an alicorn, the tables have been turned, leading to Twilight to outlive her friends, and sometimes even Spike, Luna, and Celestia.
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 whom 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 emo hairstyle 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.
Combine it with the popularity of Cupcakes and you have Pinkamena being less depressed and more Ax-Crazy.
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.
When A Canterlot Wedding aired, a lot of people were disappointed with how the inevitable fallout from Twilight's outburst at the wedding rehearsal was just casually brushed aside. Deconstructions of the episode, ranging from the rest of the cast (and occasionally Twilight as well) still feeling guilty about the whole thing to Twilight going completely nuts about it cropped up almost immediately. A subplot would usually include mention that the kingdom nearly getting conquered would serve as a black mark on Celestia and Shining Armor's reputation, especially when they were both warned beforehand and heavily prepared for a threat, usually leading to rebellions or a call for Celestia to step down, with some more darker stories having Celestia tried for treason (with or without a guilty verdict and/or an execution).
Ever since the Pinkie Pie-centric Dark FicCupcakes became a smash hit, slews of fanfics have followed in its footsteps. They typically involve a character turning evil (or turning out to be evil all along), usually for no good reason, and slaughtering one or more other characters, usually for no good reason either. Almost all Mane Six members have at least one popular fanfic with this premise. Rainbow Dash has Rainbow Factory, Twilight Sparkle has The Experiments of Twilight Sparkle, Rarity has Rarity's New Dress. Fluttershy is a variation - her Pattycakes contains no gratuitous violence, but it's often said to be just as traumatizing than the above fics, if not more so. Secondary characters get this too - Big Macintosh has the infamous Sweet Apple Massacre, Cheerilee has Cheerilee's Garden, and Trixie has Trixie's Funhouse.
A popular type of these follows the lines of "Twilight Sparkle is destined to turn evil, or is turned evil by by a malevolent force destined to re-take Equestria." These often feature torture/mind rape of the Mane Six and/or the CMC, and an epic quest to defeat her or turn her back to good.
King Sombra and his corruption are sometimes involved in this.
A seventh Element of Harmony being revealed. If an OC is involved, it usually requires a retcon of what happened when they were first used. When an OC isn't involved, Trixie is a common candidate, and with Twilight being an Alicorn, the harmony of two pegasi/unicorns/earth ponies is restored.
While not as common as the others on this page, Mr. and Mrs. Cake's having a unicorn filly and a pegasus colt has fueled more than a few investigatory fanfics.
Alternate Universe fanfics are quite popular. Either the circumstances of Rainbow Dash's first sonic rainboom (which lead to mane 6 earning their cutie marks), the beginning of Twilight's magic studies (and her relationship is Celestia) or the plot of the pilot episode (how they all became friends and defeated Nightmare Moon) play out rather differently. Alternate histories for Equestria have also been done.
Before Twilight became an alicorn, a number of alternate universe fics posed the question of how different events would be if Twilight had different attributes then usual. Either she was born as a different species, a different gender or other physical, mental or emotional attributes. A different life history where she's an orphan and gets adopted by Princess Celestia is also quite common.
What with the profusion of princesses in the setting, a very common plot involves one of the princesses holding open court and the various difficulties and adventures that result from the petitions brought before them. Became especially common upon Princess Luna's return in season two and even more so once Twilight Sparkle became an Alicorn Princess at the end of season 3. As often played for humor (ridiculous requests and royal shenanigans) as for drama (princesses angsting over responsibilities, politics, etc). Twilight-centered fics tend to wind up with both: she's extremely neurotic and a perfectionist, but just what exactly is she the princess of anyway?
The Discord/Celestia pairing shows up a lot - a common interpretation is that Discord was good until Celestia rejected him. This comic is a prominent example of this trend - even though the author himself stated that he wanted to portray Discord as an Unreliable Narrator who just treated the rejection as an excuse to embrace his true evil nature.
Due to King Sombra's lack of backstory, he gets this a lot. Many stories see him corrupted by dark magic, either accidentally or not. A lot of shipping between him and Princess Luna (as well as Chrysalis) also occurs.
Queen Chrysalis is Fetish Fuel to many and it's not very surprising that she gets shipped and Freudian Excused a lot. A common motivation has it that she just wants to feed her people, despite canon portraying her as blatantly cruel and sadistic.
Minor antagonists occasionally receive the treatment as well. Backstories for Gilda featuring Rainbow Dash shipping are particularly prominent.