We're not sure if Sony approves of this or not (we're going to guess "not"), but YouTube user FistFullofPotions has created a video showing how to transfer PSOne Classics to PlayStation Vita via PlayStation 3 - including games not included in Sony's list of the nine games "officially" available in North America.
Vita owners have to start this process by downloading the new firmware - version 1.80 - and installing it. This will allow the PS Vita to emulate PSOne games. Since the PS3 can already emulate PSOne games, downloading a game onto the console and leaving it uninstalled will make it show up in the Vita's download queue.
Once inside Content Manager on the Vita, uninstalled games will show up under the PSOne heading. This transfer method also works with PlayStation Portable and PlayStation Mini games, according to the video.
Since any games not listed in the official list have not passed Sony's approval process, the unforeseen consequences are on the user... Users at various websites have claimed success in installing 129 PS One games available on the PlayStation Network with 19 of those proving to be incompatible. You can find out more about that by reading this Polygon report.
If you are willing to assume the potential risks of doing this, check out the video here.