The popular "bloodswap", an Alternate UniverseWhat If? fic where the trolls have different blood colors than they do in canon — blood color being used to determine their place in the Fantastic Caste System and what Psychic Powers they get (if any). The most well-known bloodswap fic is Red Dead Virgo... which is actually a good example of what not to do when making a bloodswap, ex. swapping around sign- or character-specific traits and not swapping the lusii, despite the fact that the lusii and some of the trolls' traits are connected to the signs and not the blood; these are all common pitfalls of bloodswapping. Red Dead Virgo is generally pretty hated around certain circles due to starting and perpetuating said pitfalls.
There's also "kidswap" for swapping around the kids, which usually involves shifting the kids to a different family and either keeping the guardians (ex. having a Jade Egbert and Dad + Nanna Egbert) or swapping the guardians as well as the kids (ex. Jade Egbert and Grandpa Egbert).
A variant keeps the bloods of the characters the same, but reverses the social standing of the hemospectrum so that the former rustbloods are in charge and seadwellers are the lowest castes.
Perhaps the most common form of fanfic for Homestuck (or at least the most common fan-adventure), are the ones in which a completely new group of players (human or troll) plays Sburb. A Sburb session is a ready-made framework for a story and canon that the main characters weren't the only humans to start a session (though it's strongly implied that all other human sessions should be doomed to failure and Word of God has declared that the trolls were the only session for their species). Only very rarely does someone actually bother to invent their own race of aliens, despite that being far more likely canonically, and not everyone succeeds in putting a big enough spin on the premise to make it worthwhile.
Fan-Adventures consisting of the cast of other shows are also common.
Life after The Scratch is another big one. These generally revolve around the characters remembering or trying to remember their past or alternate lives. In many stories the trolls are turned human somehow.
Fake FAQs for Sburb, based off Rose's FAQ in canon. Originally, many of these fics treated Sburb and the events of Homestuck as a fictional video game that people were writing strategy guides for. Fake FAQ fics then went dormant until they were revived by the story Sburb Glitch FAQ, which eventually started a Fan Verse.
"Petstuck" AU stories, in which the trolls are animals kept as pets by rich humans, are relatively common. The majority of them begin with a normal person finding an abandoned troll (cardboard box and rainstorm optional), taking them home, gradually getting them to open up and trust humans after years of abuse by their evil former owners, realizing that trolls are actually sentient after all, and attempting to get them recognized as more than animals by society.
The few surviving trolls have to live on Earth after an assumed end of the series in which that is a thing that happens. The actual resolution of the series is generally neither shown nor described. Commonly used to set up an Innocent Cohabitation scenario that becomes not-so. The pairing is very often John/Karkat. Karkat is often shown grudgingly enjoying watching human movies with John.
The characters finishing SBURB and somehow returning to Earth, now having to live with PTSD from the horrible things they've experienced, usually focusing on the kids, specifically John and Dave.