So the official list of Vita-supported PS1 Classics is pretty barren. Shame on Sony. Fortunately, some savvy gamers have found a workaround: just use your PlayStation 3.
The above video shows you how to get games like Final Fantasy IX and Castlevania: Symphony of the Night running on your Vita by transferring them from your PS3. NeoGAF has a full list of games they've tested:
2 Xtreme
Alundra
Arc the Lad
Arc the Lad II
Arcade Hits: Shienryu
Arcade Hits: Sonic Wings Special
Bowling
Breath of Fire IV
Castlevania Chronicles
Castlevania: SotN
Cho Aniki
Cool Boarders
Coolboarders 2
Coolboarders 3
Cool Boarders 2
Cyberbots: Fullmetal Madness
Destruction Derby
Dino Crisis
Dino Crisis 2
Final Fantasy Origins
Final Fantasy IV
Final Fantasy VI
Final Fantasy VII
Final Fantasy VIII
Final Fantasy IX
Front Mission 3
Gex
Gex 3
GaiaSeed
Hot Shots Golf 2
International Track and Field
Jet Moto
Jet Moto 2
Jumping Flash!
Jumping Flash! 2
The King of Fighters '99
Klonoa
Legacy of Kain: Soul Reaver
Legend of Dragoon
MediEval
Mega Man
Mega Man 2
Mega Man 3
Mega Man 4
Metal Gear Solid
Money Idol Exchanger
Pandemonium
Parasite Eve
Parasite Eve 2
Rally Cross
Rapid Angel
Rayman
Rayman 2: The Great Escape
Resident Evil: DC
Resident Evil 2
Resident Evil 3: Nemesis
Ridge Racer 4
Saiyuki: Journey West
Silent Hill
Street Fighter Alpha 1
Street Fighter Alpha 3
Suikoden
Syphon Filter
Syphon Filter 2
Syphon Filter 3
Tekken 2
Threads of Fate
Tomb Raider
Tomb Raider 2
Tomb Raider 3
Tomb Raider: The Last Revelation
Toy Story 2
Toy Story Racer
Twisted Metal 2
Unjammer Lammy
Vagrant Story
Warhawk
Wild Arms
Wild Arms 2
Wipeout
Xenogears
Time to get your Suikoden on!
(Thanks, Illuminerdi!)