One common theme is physical injury during battles. Given Levi's role as a strong fighter in AOT, fanfictions often explore scenarios where he gets seriously injured in combat, like being wounded by Titans or in fights with other humans. This could lead to his comrades having to take care of him, creating moments of vulnerability and bonding.
Another theme is mutual understanding. Despite their different personalities, they seem to understand each other on a level that others don't. Fanfics often play on this, showing how they can communicate without words at times. For example, in some stories, Levi can tell when Hanji is about to do something reckless just by a look in her eyes. And Hanji can sense when Levi is more stressed than usual. This mutual understanding forms the basis of many emotional and plot - driving moments in the fanfics.
You can start by looking on fanfiction platforms like Archive of Our Own (AO3). It has a large collection of Attack on Titan fanfictions, including ones about Levi being hurt. Just use the search function and type in relevant keywords like 'Levi hurt' to narrow down your search.
Another one is 'Hanji's Perspective on Levi'. This fanfic is unique as it is told from Hanji's point of view. It gives readers a different look into how Hanji sees Levi, their quirks, and the special connection between them.
There could be various fanfictions. Maybe some that focus on alternative endings, like ones where certain characters survive who didn't in the original. For example, a fanfiction where Erwin survives and leads the Scouts in a different way.
One way is to check on popular fanfiction websites. Look for ones with high ratings and a lot of reviews. For example, Archive of Our Own (AO3) has a large collection of AOT fanfictions. You can search for the most popular ones there.