Since there's not one yet, I figured I'd link in some places that have aggregated and kept up to date lists of forwards compatible PSOne Classics and PSP releases. Most of the stuff is forwards compatible already, but I thought that having a place to look here in the forum when you get that sudden urge to play an old game would be nice. Save some people from wasting time looking for a game that isn't compatible. I won't be posting any lists here, for the sake of keeping the post itself short. Just links.
PSP and Minis: First up is PSVitaForum.com's PSP and Minis compatibility lists. The forum post includes lists of Sony Confirmed PSP and Mins games, as well as a list of games that aren't officially recognized by Sony but do work via transferring them to the system.
PSOne Classics: Second is a pair of sites with overlapping results. VG247.com's PSOne Classics EU and NA List and NeoGAF user KalBalBoa's apparently NA only Compatibility List. VG247s list includes some games that the NeoGAF post doesn't, and the NeoGAF post includes some that VG247 didn't, so I decided to include both. The NeoGAF post also includes a tutorial for transferring games that don't give you a Vita install bubble on the media bar when you purchase them and a list of games that currently aren't compatible. (I don't know how long it has been since the NeoGAF post was updated, so some of the games on the incompatible list might be working now. I'd suggest approaching any of them with caution, however.)
PS2 Classics: Sony hasn't currently announced plans to port any of the PS2 Classics offered in the Playstation Store to Vita, probably because they are making bank off them in HD bundles. However, with new consoles supposedly around the corner and Vita sales slow I wouldn't be surprised if part of their E3 conference next year, along with the announcement of the PS4, is the arrival of PS2 Classics to the Vita's stable. The system certainly seems capable of handling those games so unless there's a serious software format hiccup I'm not familiar with it would appear to be only a matter of time. If, and hopefully when, they patch in PS2 support I will be sure to update the thread.