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Sans and Papyrus are siblings, and are encountered together throughout events in the Snowdin region; Sans often has guest appearances in Papyrus' phone calls. When the two are together, Sans is often depicted annoying Papyrus with bad skeleton jokes and puns, but whenever Papyrus is offscreen Sans brags to the player about how cool Papyrus is. Missable events and dialogue also depict Sans as supportive of Papyrus' dreams and interests. Their house features a post-it note conversation involving Papyrus nagging Sans to clean and Sans being deliberately obtuse to annoy him. Many fans have noted how realistic a sibling relationship they share.
If Papyrus is killed on a neutral route, Sans stops appearing until the very end of the game, and is extra hostile during the judgment scene and end-of-game phone call.
Due to their closeness, the mystery involving the two's origins, and fandom's typical characterization of Sans' angst, Sans and Papyrus' relationship is often interpreted as verging-on-unhealthily close ride or die-esque, "we are the most important thing to each other and only we can truly understand each other". As this type of codependency is heavily romanticized (both in the sense of "idealized" and "seen as romantic") in fiction, and due to the allure of forbidden love tropes, this naturally led to many fans shipping them, and Sans/Papyrus is one of the most popular ships in Undertale fandom, both in Western circles and internationally.
However, due to the growing prevalence of purity culture in fandom overall and the number of anti-shippers in Undertale fandom in specific, Sans/Papyrus shippers are common targets of bullying and harassment. The rhetoric is generally that incest is icky and bad and that if people write about it then that must mean that they think real-life incest is okay. (This ignores that a common trope in Sans/Papyrus fics where W.D. Gaster is portrayed as their father or creator is to have Gaster abuse one or both of them, often sexually, and use that trauma as a source of angst.) A Kotaku editorial complaining of Undertale fandom's "toxicity" even went so far as to list Sans/Papyrus shippers as a problem of the same magnitude as interfandom harassment, despite that shippers are often victims of that very same harassment.
As with gen fics involving the skeleton brothers, Sans/Papyrus fics often feature Sans' angst about his mysterious backstory and grief for/overprotectiveness towards Papyrus, though Papyrus fans tend to portray the two's dynamic as more equal. The aforementioned abuse by Gaster is another popular element. AU fics often have Sanses and Papyruses from alternate universes hook up, or compare and contrast the relationships of different sets of the characters.
Sans/Papyrus is the third most popular Undertale ship on AO3.