I love Borges's many short, short stories. Many of them, I assume, are under a thousand words. Other than that, I can't recall any really, really short stories that have stuck with me, that have shaped who I am in any significant way. I wrote one, just to see if I can do it. I guess it was on okay story, but even I left it feeling like I had just sketched out something else I needed to write later as a longer story.
I think flash fiction is a bullshit genre. It's everywhere now, and editors and writers both are swearing that you it can carry the full power of longer fiction, but I haven't seen examples. Flash fiction (and the "i-story") are just bad attempts to make fiction seem to be keeping up with the times, these go-go times of short attention spans and four social media running at once. The result seems to me to be embarrassing, like when churches try to ape pop culture to seem hip.
Flash fiction often seems like an easier challenge than the "six word story" challenge. It's an etude, a thing for writers to try, and they may learn something from it that will allow them to write smarter when they get back to "real" stories.
The only thing I do like about the short-short is that it tends to be lenient about allowing authorial intrusions, which I think modern lit lacks. An example is Sherman Alexie's "Idolatry," which allows for this country-westernish moral to be drawn at the end: "In this world, we must love the liars. Or live alone." You'd never get away with that in longer stories. We accept it in short stuff, because we are specifically looking for a novel-sized life lesson in a pill, and so we allow it. It's like Jesus telling a little vignette, and then wrapping it up with "So I say unto you..."
I'll probably write more of it, though. Just like at my job, I hate writing my own performance review, but eventually started putting effort into it because I was tired of mediocre reviews, I will probably also work harder at making flash fiction because I want to get stories told, even if they're not really in the form I would prefer to write them.