Oh boy, this is where things get wild. The pure-blood families are basically treated like magical royalty in so many fics, with all the scheming and arranged marriages you'd expect from a medieval court, only with wands. You'll see a ton of stories where the Blacks, Malfoys, and Potters are actually part of some hidden sovereign wizarding council, or where the 'Sacred Twenty-Eight' families have secret treaties and ancient obligations that dictate everything. The trope of Harry being the 'Lord of multiple Ancient and Noble Houses' is practically its own genre, complete with inheritance tests, political alliances, and enemies trying to sabotage his claim. It's all about taking the classism that's baked into canon and turning it into a full-blown power structure. I have a soft spot for the ones where the intrigue is subtle, like Narcissa Malfoy maneuvering through social events to gather information, rather than just heavy-handed lordship edicts.
A more specific common plot is the 'lost heir' scenario, where someone like Hermione or even a muggle-born character discovers they're actually the last descendant of a royal magical line that was thought extinct, usually Slytherin or Gryffindor. This immediately throws the existing 'royal' hierarchy into chaos. The conflict often comes from the established families trying to either eliminate the new claimant or co-opt them through marriage. It’s a convenient way to give an outsider character immense social power while keeping all the drama of courtly politics.