Before the series came out, Ruby and Weiss were shipped together constantly. This was most likely caused by the DeviantArt author LunarisFuryAileron, whose RWBY4komas were featured on Monty's Twitter a few times. This continued on even after Ruby and Weiss were revealed to have very different characterizations than what the fandom assumed.
Blake and Yang shared almost no interaction before they became teammates halfway through Volume 1, but that hasn't stopped the so-called Bumblebee pairing from being one of the largest pairings in the show. Fans were delighted when the show gave them a little heart-to-heart talk in Volume 2 and a more tragic incident at the end of Volume 3.
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.
Roman Torchwick and Neopolitan are shipped very often by the fans because of the fact they are predominately seen in each other's company and are both extremely psychotic. Roman's death has not hurt the ship as much as Roman calling out Neo's name when she's blown off the ship has helped it.
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 This Is It 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 two most popular within the fandom are Vax/Keyleth and Vex/Percy. It helps that the first is practically canon, the second has a fair amount of Ship Tease, and neither ship conflicts with the other.