There is no single definitive list covering **every** PS2 game's compatibility on PS3, but here is how to determine what works:
Methods of Play
PS2 Classics (Digital): Sony re-released select titles on the PS3 PlayStation Store. These are emulated, trophy-supported, and guaranteed to work. Check the Store for the full current catalog.
Hardware Backward Compatibility (Disc): Only launch-model "Fat" PS3s (CECHA/B/C/E, 20/60/80GB) have the Emotion Engine chip for near-full hardware compatibility. Later "Fat" models (CECHG/H/K) used partial software emulation (lower compatibility). "Slim" and "Super Slim" models cannot play PS2 discs.
Best Resources for Checking Specific Titles
PS3Trophies.org / PSNProfiles.com: Community-maintained compatibility lists for disc-based play.
Wikipedia "List of PlayStation 2 games compatible with PlayStation 3": Good reference for software-emulated models.
Reddit (r/PS3, r/emulation): Search for specific game titles + "PS3 compatibility".
Important Caveats
Disc compatibility varies by PS3 model; always verify your specific model number (CECH-XXXX).
PS2 Classics are region-locked to your PSN account region.
Online multiplayer for PS2 games is non-functional (GameSpy shutdown).