In canon, Weiss and Neptune are attracted to each other, however the fandom prefers Weiss with Ruby and has given the ship a nickname ("White Rose"), having shipped them together ever since their prequel trailers first came out. The fandom imagined a pairing between The Stoic (Ruby) and a Tsundere Weiss. However, the proper show revealed the two girls had very different personalities than assumed. This didn't dampen the ship, but did prevent the canon ship from developing massive support within the fandom.
In canon, Blake and Sun are attracted to each other, however the fandom prefers Blake with Yang and has given the ship a nickname ("Bumblebee"). This again originates from the prequel trailers, with the canon show giving the girls different personalities to what was initially assumed. Blake and Yang don't meet until half way through the first volume, and the fans delight in every emotional scene between them. Bumblebee is the most popular ship pairing in the entire show.
But whether this shipping isn't canon is yet to be determined. Sun has definitely pursued Blake, and she was warm feelings for him, but she also has warm feelings for Yang and Yang for her. Whether or not Blake and Yang have a romantic or sexual attraction, Weiss points out in later seasons that Bumblebee is one of the closest friendships within the team, as close as or possibly closer than the half-siblings Yang and Ruby. When the party gets split for a season, even though Yang goes looking for Ruby she appears more distraught over being away from Blake.
There are fan names for most or all of the combinatorial shipping possibilities, even the improbable ones like Qrow and Winter. Sometimes the fans have multiple names for the same ship; the RWBY fanbase really likes shipping and will defend their ships with great fervor. Some shipping names for pairings within team RWBY got a surprise official blessing, when the show writers used them as Ruby's names for team RWBY pair tactics during a battle with a giant mech.
Sun Wukong gets shipped with his Casanova Wannabe team mate Neptune, which initially began based on Neptune's appearance alone. This later gained popularity as a result of his adorkable personality, as well as a desire to solve the perceived threat to both the Bumblebee and White Rose ships which arises from the Ship Tease between Sun and Blake and Neptune and Weiss.
Enough comments appeared on the Worm threads shipping Skitter and Clockblocker together that the author had Clockblocker explicitly sink it in canon.
Beyond Skitter/Clockblocker, other pairings like Skitter/Panacea and Skitter/Tattletale tend to be much more popular than canon pairing Skitter/Grue.
It would probably be easier to name the Don't Hug Me I'm Scared fanfics and fanart that don't ship the Notebook and Tony the Talking Clock. The two of them have never interacted, and they've never been onscreen at the same time for more than a few seconds.
Nyan~ Neko Sugar Girls fans overwhelmingly ship Koneko and Raku despite the fact Raku pines for Hitoshi the entire series. The finale where Raku dies in Koneko's arms as she cries only cemented the pairing. Prior to that Kidnapper-kun/Hitoshi was the most popular ship until it became the Official Couple.
For most of the Red vs. Blue history, Grif/Simmons has been the main ship, which is helped by the large amount of Ho Yay and creators implying it's canon.
On the Yuri side, Agents Connecticut and South Dakota, who never had a single meaningful interaction onscreen.
As of Season 11, Tucker/Washington has become one of the most popular Red vs. Blue ships.
Critical Role has its fair share of ships, but the most popular by far is Percy/Vex'ahlia, which outstreches even its own Ship Mate turned Official Couple Keyleth/Vax by a wide margin. This is likely thanks to the sheer amount of long drawn-out Ship Tease that's been in place since the Briarwoods arc.
There's a large amount of Ruby Quest fan-art about Stitches/Daisy, who never met in the game and whose only connection was both being met as horrible mutants. It may be a case of Pair the Spares, as they are the only patients of the Metal Glen not dead ( Maddie) already in a relationship ( Ruby and Tom, and Tom Nook and Lucy), or escaped without a romantic partner ( Jay).