This is a weirdly specific cross-section of the fandom and I'm kind of living for it. The classic 'Joke' ship is obviously Ms. Joke/Aizawa, that's the whole point of her character. Some writers really lean into the potential awkwardness of a professional hero with a pun-based quirk trying to thaw the eternal grump, and it can be fun. But honestly? I've read a few where Aizawa is paired with Present Mic and Ms. Joke becomes this amazing, supportive third wheel who just shows up to annoy them both, and it's unexpectedly wholesome.
On a weirder note, I stumbled across a 'crack' fic ages ago that shipped Ms. Joke with Mirko. The logic was something about them both having loud, high-energy personalities, but Mirko's being pure aggression and Joke's being performative. It was surprisingly well-written, exploring how two people who seem similar on the surface could have completely different motivations.
I also have a soft spot for fics where Ms. Joke mentors a younger, more anxious student—sometimes it's a younger version of herself, sometimes it's an OC—and Aizawa begrudgingly gets dragged into it because the kid shows potential for underground hero work. It's less a romance ship and more a found-family dynamic, but the slow-burn respect between the two pros is the real draw.
Most of the good stuff I've found tends to be on AO3, filtered by the 'Ms. Joke & Aizawa Shouta' relationship tag, then digging through the related works. You find the real gems in the bookmarks of authors who write that dynamic well.