Kindle Singles are made possible by the simple fact that ebooks don't have the same distribution costs as paper books. The reason why many novels are between 200 and 400 pages is not because it is ideal for story telling but because the printing and distribution costs make novels outside that range uneconomical.
Amazon launched the Kindle Singles in 2011, over 3 years after the initial launch of the Kindle Store, but the idea is actually much older than that. While I cannot vouch for other ebookstores, I know that Fictionwise was selling short fiction since at least 2006. That's when I bought it, at least, and I'm sure that they had been carrying those works for some years beforehand.
While Amazon is pushing to sell nonfiction content as Kindle Singles, I seem to recall that most of short works at Fictionwise were fiction. Of course, that could be a result of the ebook market being biased in the favor of fiction content during those years, or it could be that I simply didn't look for other content.
The prices I saw at Fictionwise ranged from $.49 to $.199, and all of the titles were DRM-free. I built a large library of short story collections from various authors, including Silverberg, Asimov, Resnick, and more.
Those short stories and novellas are usually sold (in paper form) as anthologies, but thanks to the minimal distribution cost of ebooks the authors were able to to sell the stories individually.