Common themes are transformation, not just in the sense of the Miraculous transformations but also in the characters' personalities. Hawk Moth might transform from a pure - evil figure to something more complex. Another is the exploration of the Miraculous powers. Fanfics often dig deeper into how the powers work and how they can be used in different ways against Hawk Moth. And secrecy is a big theme. Keeping identities secret, Hawk Moth's secrets, and how those secrets impact the story.
One great fanfiction is 'The Hawk Moth's Redemption'. It explores an alternate universe where Hawk Moth starts to question his motives. Another interesting one is 'Ladybug and Hawk Moth's Unlikely Alliance' which has a unique plot of the two having to team up for a common cause. And 'Hawk Moth's Hidden Past' delves into his backstory and how he became the villain we know.
It could be Gabriel Agreste. He has been a suspect in the original series for various reasons. He is often absent and has a lot of secrets. So in the fanfiction, it would make sense if he was finally revealed as Hawk Moth.
There could be fanfictions where Gabriel as Hawk Moth teams up with unlikely allies. Perhaps he realizes that his way of doing things is wrong and joins forces with Ladybug and Chat Noir to fight a greater evil that threatens the entire city or even the world.
I'm not sure of specific titles off the top of my head, but you can check out fanfiction websites like Archive of Our Own. There are likely many stories there that focus on Gabriel as Hawk Moth. You can look for ones with high ratings and lots of reviews.
One possible plot could be that Ladybug and Chat Noir finally track down Hawk Moth's lair. They discover that Hawk Moth is someone they know in their daily lives, like a teacher or a neighbor. This revelation shocks them, but also gives them an edge in the fight as they now understand his motives better.
In an akumatized Ladybug fanfiction, the akuma might enhance her powers in unexpected ways. For example, her Lucky Charm could become a more powerful and unpredictable object. Her akumatized form might also have a darker and more menacing look, perhaps with a more aggressive fighting style. Writers could explore how the heroes and villains interact with this changed Ladybug.
In this 'Miraculous Ladybug Criminal Ladybug Fanfiction', there could be a lot of potential storylines. One possibility is that an external force is influencing Ladybug to commit crimes. It could be a new villain who has found a way to manipulate the Miraculous. This would create a great conflict as Ladybug has always been the symbol of good. Her friends and allies would be in shock and would have to find a way to stop her without hurting her. The story could also explore her internal struggle, as she might be aware of what she's doing but unable to stop herself.