**Okay, so I started this list last week and kept it on private while I built it and added links... and then completely forgot to unlock it, so lemme try this again.
I've heard this type of fic referred to as safety fic/safety tabs, cuddle!fic, and anti-angst fic (though it doesn't mean that the stories do not contain angst, simply that by the end of reading them you feel content). It's your fandom version of a fuzzy blanket, a cup of hot tea, and that favorite novel you've reread a million times.
This is far from an exhaustive list (I haven't even started slogging through my Harry Potter links, for example). For now I'm just going to focus on the pieces that I tend to turn to for a reliable pick-me-up when needed.
Stargate Atlantis:
The
Farm in Iowa series by
sheafrotherdon. (John/Rodney, various ratings through NC-17, AU, kidfic)
The
Nantucket series by
aesc,
dogeared,
sheafrotherdon, and
siriaeve. (John/Rodney, various ratings through NC-17, AU)
The
On Life and Living series by
wesleysgirl. (John/Rodney, various ratings through NC-17, AU, kidfic)
A Supermarket in California by
taffetablue. (John/Rodney, R, AU)
People Watching by Tipper (link to fanfiction.net). (Team(ish) mostly featuring Carson and Elizabeth, PG)
Star Trek (yeah, it's new incarnation is relatively young in fandom terms, but I've still found some good safeties!):
The
Village!verse by
chaletian. (Ensemble, PG through PG-13)
Two Men & a Motorbike series by
inell (Jim/Bones, rated teen through adult)
Hard-Earned Rights by
andrealyn. (Jim/Bones, PG-13, kidfic(ish))
How to Avoid Kicking Puppies and Other Valuable Lessons in Leadership by
sparky77. (Ensemble, R)
This ficlet by
theonlytwin. (Spock/Uhura, Spock/Uhura/Kirk, R(ish))
Discovered Check by waketosleep (links to dreamwidth). (Spock and Kirk preslash, G)
Friends: A Kirk and Bones fanvid (links to Youtube)
We Go Together fanvid. I never would have imagined putting Star Trek and Grease together, but ya know what? It totally works!
Torchwood:
Jack's Promise fanvid (Jack/Ianto)(links to Youtube)
Firefly/Serenity:
Defying Gravity fanvid (links to Youtube)
***I have no explanation for why so many of my safety fic are AUs and kidfics. I JUST DON'T KNOW. This is not normal for me, really, because I started out very gen-focused and character exploration driven. Now it's all about the slash and exploring different worlds/dimensions/timelines/possibilities, etc.