The sheer volume of 'Dungeon ni Deai wo Motomeru no wa Machigatteiru Darou ka' fic out there means Bell and Aiz have spawned some seriously defined patterns. The most classic has to be the timeline divergence where Bell doesn't join the Hestia Familia—maybe Hestia isn't there that day, or he bumps into Aiz first. Those stories often explore him joining the Loki Familia as a mascot-level rookie, which creates this fascinating dynamic of him being trained directly by her from day one. It's a great way to force their proximity and bypass the whole 'unattainable idol' phase of their canon relationship, letting the mentorship and eventual romance develop faster. Another massive one is the 'what if Bell's skill was different' trope. Instead of 'Realis Phrase' scaling with his feelings for Aiz, maybe he gets something tied to wind or spirits, creating a more direct magical link to her that she can't ignore. It's a neat engine for plot.
Beyond the big plot changers, there are tons of softer, character-focused tropes. I'm a sucker for the ones that involve Aiz dealing with her own emotional repression. There's a whole subgenre of fics where she accidentally overhears Bell talking about her with pure admiration, not just as 'the Sword Princess' but as a person, and it cracks her stoic exterior in a way the canon hasn't yet. Those little moments of her trying to understand human connection, maybe asking Riveria for bizarrely phrased advice, are gold. Also, you can't ignore the hurt/comfort standby. Bell gets injured saving someone from the Loki Familia, or Aiz takes a bad hit protecting him, and the aftermath forces a vulnerability and care-taking that neither is equipped to handle gracefully. It's formulaic, sure, but when done with attention to their characters, it hits the spot.
I'd also toss in the 'Aiz gets temporarily de-aged or loses her memory' trope. It's a bit out there, but it forces Bell into a protector role while a more childlike, unburdened version of Aiz gets to experience simple joys, which often makes the restored Aiz reevaluate her life. It's a shortcut for character development, but it's fun. The common thread in all these, I think, is a desire to bridge the massive gap in power and status between them faster than the source material can, to get them on a more equal footing where a real relationship could feasibly bloom. Sometimes you just want to see the payoff without waiting ten more light novel volumes, you know?