These are the plots I've often seen in Loud House fanfiction. Other people have made similar lists, but haven't listed some of these, and I will try to provide as much information as I can.
Runaway Lincoln[]
This is one of the most infamous plots and has been parodied several times. Basically, it involves Lincoln feeling sad and/or angry, usually related to something one, more, or all of his sisters did (possibly their actions in "No Such Luck"), or the chaotic life in general, or feeling that they don't love him, and deciding to run away from home and live somewhere else.
This "somewhere else" varies, but it could be the McBride house, the Santiago house (if set/written before the Santiagos moved in with the Casagrandes), or nowhere in particular.
While Lincoln has wanted a break sometimes, if he did run away, he'd likely come back of his own volition, since his breaks aren't usually more than a few hours at most. Additionally, Runaway Lincoln stories tend to have Lincoln be a lot more emotional and thin-skinned than he is in canon, sometimes even whiny, and for his affection for his sisters to be thrown out the window. While he has done selfish things in the past, it's shown that he does care about them. Even "Kings of the Con", which is maligned for him being selfish in it, has him realise his sisters deserved the prize and give it to them.
Runaway Lincoln stories are usually resolved by his sisters and parents finding him and/or convincing him that they love him.
Sequels and rewrites to episodes[]
This is not bad. I have seen perfectly good sequels and rewrites of episodes. That said, there are a fair share of bad ones too.
A sequel is a fanfiction that takes place after the episode, as a follow-up, while a rewrite is taking an episode and changing the plot a little (or a lot).
Both are often because a writer dislikes, or even outright hates, an episode, such as the aforementioned Runaway Lincoln stories based on "No Such Luck" (and they're writing the fanfic either as a "coping mechanism" or to give the epsiode a sense of completion), though rewrites may be written for different reasons, such as versions of the episode set in the gender-swapped dimension or the rabbit dimension, or things such as "what if it'd been so-and-so who'd done X instead of so-and-so in Episode Y?".
Sequels to disliked episodes usually involve a character who behaved in the way the writer disliked apologising for it, being told off for it, feeling very guilty for what they did, or otherwise getting "karma" for it. Rewrites of disliked episodes usually involve characters behaving more sensibly than they did in canon (for instance, a rewrite of "Brawl in the Family" might involve Lori and Leni fighting for a different reason).
A common trend in "No Such Luck" sequels and rewrites is the idea that the family (minus Lynn) were only pretending to think Lincoln was bad luck as a punishment for lying. The sequels with this plot have Lincoln finding out it was fake later, and the rewrites take place during the episode but with the audience knowing that it's fake.
Making characters act nicer or meaner than in canon[]
One cliche that was common when the show first came out but less common nowadays is the "nice sisters, mean sisters" concept which involves the idea that Leni, Luna, Lucy, Lana, and Lily are the nice sisters, while the rest are the mean sisters. This probably stems from Leni being generally kind, Luna being nice to Lincoln in episodes like "For Bros About to Rock", Lucy being quiet (or possibly her being sensitive in "Sleuth or Consequences", or a combination), Lana being politer than her twin, and Lily being a cute baby, and on the flip side Lori's bossiness and occasional short temper, Luan's April Fool's behaviour, Lynn's competitiveness and roughhousing, Lola's brattiness, and Lisa's tendency to experiment on her family.
Even without the "nice sisters, mean sisters" thing being in full effect, characters in fanfiction can still be guilty of acting uncharacteristically nice or mean, usually the later. This may come from a writer's opinion on a character, but may also be for the sake of plot.
Uncharacteristic meanness is often caused by Luan's pranks turning full-blown evil (I know you may bring up the bleach, the billboard, the bed, etc but, through cartoon logic, the Louds didn't seem to be in any real danger then), Lori, Lynn, or Lola being even ruder than in canon, or all of Lincoln's sisters ganging up on him and spiting him mercilessly, often for longer than they would in canon. Sometimes, Lincoln is shown being uncharacteristically mean to get "revenge" on his sisters, and he often gets away with this, and sometimes the parents' tendency to not intervene is blown out of proportion or they unfairly punish their kids.
Uncharacteristic niceness is usually present by someone, usually Leni, Luna, or Lucy, not taking part during a sibling argument scene. While they are nice girls, most sibling argument scenes (e.g. "Slice of Life", "It's a Loud, Loud, Loud, Loud House") have all the siblings taking place. Also, there's the idea that Luna is Lincoln's closest sister, with some even calling her his "guardian".
Dark explanations for things[]
These are fanfics written by writers who want to know why something happens. Why does Luan go crazy on April Fool's Day? Why is Leni dumb? Why does Lincoln have white hair? And they come up with a reason that is a tearjerker or scary.
Maybe Luan goes crazy because she's literally psychotic, or Leni is dumb because she got brain-damaged as a baby, or Lincoln's hair (or Lucy's skin) is white because of a strange disease. It may involve the parents or the older sisters explaining to the younger siblings.
Romance[]
Whether it's between the canon couples (like Lobby, Lubenny, and Saluna), non-canon couples that make sense (like Clyde/Haiku or David/Darcy), non-canon couples that make a little less sense (Luaggie is an example since Luan was only a clown at Maggie's party), or non-canon couples that are outright depraved (such as the infamous incest and pedophilia fanfics).
A common romance plot is Clyde getting a new crush on one of the older Loud sisters after dropping his crush on Lori. The new crush is often Leni or Lynn.
There's also the idea of Lori and Bobby becoming teenage parents, or Lily being actually Lori or Leni's daughter.
Serious injuries/death[]
Fanfics that involve a character dying, being at death's door, or becoming disabled. Some of these stories are written by authors who desire the actual show to be darker than it is, but that's not a necessary condition.
Lincoln is a common one to either get a terminal illness, become injured and the family worries that he will die, or, less commonly, try to kill himself. Lucy is more common for the "trying to kill self" plot. For added angst, one of the sisters will often feel responsible somehow. Maybe Luan injured Lincoln in one of her pranks or Lynn during sports.
Lisa having a near-death experience due to an experiment or invention gone wrong is slightly less common but still a thing. Lynn being confined to a wheelchair and/or having an amputation is also common because it would affect her ability to play sports.
Lucy or Lisa causing something weird[]
Want the Louds to turn into animals or babies or whatever? Need some time travel? Want a story about ghosts or alternate dimensions? Simple-- just have Lucy do some (often accidental) magic or Lisa either invent something or have an experiment go wrong that causes the change. These stories are usually played for comedy.