I can't speak to the more general military history of the Sling but I can say that it was in use at least as late as the Battle of Falkirk in 1298. We have records of the summons ordered by Edward I of England for that campaign and there is an entry on somewhere sending slingers as their contribution to the campaign. In my mind the most amusing part of all of this is that the place that sent them was the territory around Sherwood Forest, mythical home of Robin Hood.

Whether this was entirely an odd anomaly or evidence for the continued use of the sling in warfare I can't really say. I can't even say if they actually made it to the battle of Falkirk and/or made any noteworthy contribution. There is evidence that they showed up at the muster for the army but that's about it and they certainly don't feature prominently in discussions of how the battle actually unfolded.

Prestwich, Michael, War, Politics, and Finance under Edward I, (London, 1972).

It depends to some extent on whether you mean the traditional shepherd's sling or the staff-sling (or "fustibal").

The Spanish were, as far as I know, the last people to make significant use of traditional shepherd's slings. They continued to use them at least into the 14th century. There were a significant number of slingers, for example, at the Battle of Najera.

The staff-sling held on longer, being used to throw grenades into about the 17th century. I'm having trouble finding exactly when staff-slings became unpopular as a way to throw grenades.

Sometimes, in desperate circumstances, slings could be used later. In the Spanish Civil War, there were some occasions where slings were still used to throw grenades (so far, the sources I can find are all accounts of Republicans using slings, not Nationalists).

(I've read personal anecdotes of usages that occurred later, but the story I had in mind took place within the last 20 years, and at any rate my source is some informal correspondence with an alleged veteran over the internet whose real identity I can't verify, so I guess I won't post that.)

Slings (and, moreso, slingshots) were used in the First Intifada (possibly as more of a propaganda/"fourth-generation war" tactic than for their military effectiveness), but they're still seen in Palestinian/IDF clashes today.