Last night, Sony graciously made nine whole PSOne games available for North American PS Vita users to download. However, gamers soon discovered that you could transfer a number of games over from the PlayStation 3, greatly expanding the potential library.
Sony hasn't officially acknowledged this method and it's not exactly intuitive, but once you know how, it's simple enough -- not as simple as it should be, of course, but we work with what we have. In any case, here's a quick step-by-step guide on how to get your PSOne Classics to your Vita.
It's worth reiterating that not all titles are confirmed as transferable, though a great deal of them are. Unfortunately, you will have to look online or find out by trial-and-error. Have fun experimenting ... rather than have fun playing videogames.
Games currently confirmed incompatible (as noted on GAF):