Sony may have included the PS2's CPU and GPU with every PlayStation 3 system, but the system still appears to be having some problems with backwards compatibility. Just in time for the PS3's second anniversary (the second day it's been on the market in Japan), Sony passed on the specifics.
Now accessible at the Japanese PlayStation.com site is a page for seeking out the backwards compatibility status of PS2 and PS1 titles. The Japanese page can be accessed here. By inputting the title, publisher name, release date or product number (an eight digit code containing four letters followed by a hyphen then four numbers), you can seek out the status of the 7,841 titles in Sony's database.
A warning about this search engine. It's a bit tough to read, as it lists all versions of individual games separately. A game might have its main version listed separate from its limited edition, its budget version, its second budget version, and so forth. This is one of the reason you'll find over 350 games listed under Konami, although Konami hasn't actually released that many games.
Most of the titles have the following message next to them "Currently, there have been no reports of major problems during play." Whether or not the games actually don't have a problem, or there is a problem but Sony hasn't found it yet, is up in the air.
IGN did find a few problem titles by looking through the list. Here's a quick run down.