Shippers gonna ship. Have you seen the RWBY FNDM? They've pretty much shipped everyone with... everyone and everything.
The popular character would get paired with whomever the fans want to ship them with. And if they can't find a decent ship, they'll make an OC and ship them. Worst case scenario, they'll form a ship with an established pair, and want the hypotenuse to die for their ship.
You remember that Lorax movie? Fans were drawn to the Once-ler character and wanted to ship him with other characters but there weren't any other characters. So fans shipped him with himself. Like other versions of that same character. It was very strange.
Well, I do like pancakes
Just sorta make your character conventionally attractive (by your audience's standards) and have charisma.
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The example of the Onceler is a very good one actually. I wouldn't be worried for you. If you make the character attractive and charming, it'll work.
I think that this would be an interesting social experiment to test what fans would do. I would write a few fanfics where your character is shipped with someone from another fandom (Probably one that's obscure or one that doesn't really go well with your story). It would be interesting to see if actual fans get behind that ship just so they can have something to ship.
I expect they'd end up shipping your character with themselves. Which is really what shippers tend to do anyway.
This is a thought experiment I came up with, as I know my works probably won't ever get popular enough to have shippers. What if an author created a character that the fans would be drawn to, that they would desperately want to ship with other characters, but then didn't give the fans someone to ship that character with? What would happen? How would someone go about doing that?