Do the one-sheet Savage Worlds adventures only require the core rules, or do they generally use some setting-specific rules?
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Just the core rules is fine. For that matter you could probably get away with using the Test Drive, given they also provide pre made character downloads.
You might need the setting supplements for fluff information, depending on how detailed you want to run it, but that really shouldn't be necessary. I always used to run adventures from Dungeon which I didn't have the campaign setting for.
At one point, Pinnacle posted submission guidelines for one sheets which explicitly stated that they should never require anything beyond the core rules, even if written for a setting. Unfortunately, it seems like they have taken them down so I can't refer you to them, but I imagine that they are still following this policy. Thus every official one sheet is required by Pinnacle to only require the core rules.
The only exception they made was if a one sheet was written for Necessary Evil or the Super Powers Companion in which case the new superpowers rules introduced in them could be required. To date, there are no official one sheets written for this though, so that shouldn't be an issue.
There is actually unfortunately some need for non-core rules in some one-sheets.
For example, Smuggler's Song indicates that the Watch Sergeant has the Musketeer Edge. This is not explained in the core rules…