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Whatever app that probably has a share sheet (apps you can email or message from)
yes, I am sure more apps will take advantage of this, but Apple's apps use it (quick list)

Safari
Maps
iTunes Radio stations
Contacts
Notes
Podcast stations
Voice Memos

also:
NASA's Apod app (click Share button>Save or Share> there it is.)

possibly more, that's just a quick list.
 
I think Airdrop has it's own API so other apps support it. When Apple launched the GM and allowed app submissions they referred to Airdrop so that will be cool.
Btw some apps already show the new share pop up if you hold on top of pictures to save, for example. So I guess it also depends on the type of file, but I'm not sure how it works.
 
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