I can currently do this using Dropbox, but it's rather a hassle; the quickest method I've found is to use the web client[1], navigate to my Public folder, drop a file onto it, wait for it to upload, then right-click -> Copy Public Link.
Making that one step would be wonderful. Making the link a "promise"[2], so I don't have to do synchronous clipboard gymnastics when I'm trying to get multiple links at a time, would be even better.
Droplr (https://droplr.com/hello) is pretty similar to all this--but it doesn't do direct links, so my most common usage-pattern for it (treating it as an "I need a CDN to show people this image/audio/video for the next 30 seconds or so, then it can disappear" service) is mooted. Dropbox also changed their service to remove Public folders for new users, with the "shared files" in its place having the same wrapper+HTML embedded file design.
Temporary hotlinking of my own content is important; I'm willing to pay for it!
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[1] The Dropbox native client's uploading thread is optimized for background "forget about it, it'll just be magically synced later" processing, not synchronous use. The web client lets you upload as fast as your web-browser can manage.
[2] A "promise" link would be something like--if the file hasn't finished uploading yet, figure out whether the User Agent is a human or a script/bot. If it's a script/bot, just let their connection sit open until the file finishes (they won't mind.) If the user-agent is human, however, immediately return a 408 (Timeout) error with HTML body content showing a spinner. Estimate time-to-completion plus a little bit, and put that time into an HTTP "Refresh" header. (Remember those?)