The American version of the PlayStation 3 backwards compatibility database is now online, allowing you to search through hundreds of PS2 and PSone games and see if they work on Sony's new console.
All you need to do to see if a game works is type in its name (e.g. "ICO") and hit Submit. There are additional search filters for publisher and platform, and if you get a hit the page lists the game by title, with publisher and product number, and a description of its current state (e.g. "Throughout gameplay and FMAs, the screen flickers.")
Sony has previously admitted that the console doesn't run about 200 of back-catalogue titles, although the company hopes to improve the situation in the near future through firmware updates - with some previously broken titles (notably in Japan) already implemented via the 1.10 upgrade.
The good news, such that it is, for European PS3 consumers is that by the time the console materialises here next March, the situation should be much improved.