I've written a great deal of flash fiction, < 1000 word stories. I know I can send them out to literary journals and online literary journals, but are there paying markets for this sort of thing?
There are very few, if any, paying markets for Flash Fiction. There's not really a large market for readers of flash fiction, so it's not worth the investment for most places to pay for them. A lot of authors usually only write flash fiction with the intent to give it away for free. It's a marketing tool. It gets the author's name out there and can get them readers for their paying fiction.
There are quite a few paying markets for speculative flash fiction. Fewer for mainstream. And when I say "paying," that can mean as little as $10. However, look around and you'll find plenty of markets. Here's Absolute Write's list, but it's not all inclusive.
I'm not familiar with markets that specialize in flash fiction, though these may exist. However, if you feel you've got a worthwhile, marketable story, I think most venues are happy to consider flash fiction submissions. So if there's a market that would be appropriate for a story "if it were longer," it might be appropriate as-is, as well.
I've seen stories that could be described as flash fiction in, say, The Magazine of Fantasy & Science Fiction. Flash fiction has a certain advantage in that it's short, and can be used as a sort of "filler," making it easier to juggle page count and not incurring much cost to bring aboard. On the other hand, as Ralph touched upon, many readers find flash fiction unsatisfying and uninteresting, and there may be a certain glut because (presumably) it's easy to churn out a lot of mediocre flash. My guess would be that in total, if it's very good you can probably sell it (eventually), and if it isn't then you probably can't. Just like pretty much any story :)
In my experience with self-publishing e-books, I have found that a lot of authors will use flash fiction or short stories as freebies to encourage people to buy their longer works. I also know several people who will sell their self-published flash fiction and short stories at a very low price, and then sell larger collections as the real breadwinners.