I have an HP g4-1387, core I5, 4Gb RAM, 8Gb swap and hybrid Intel/ATI graphics with Ubuntu 12.04.1 LTS.
Finally after installing AMD Catalyst 12.4 as described here I was able to Suspend. Hibernate and shutdown. With any other version superior to 12.4 my system hangs on shutdown, hibernating and suspending. Now it works but I am not able to hibernate correctly. When I do it I see disk activity "saving" my data, but when I turn on my computer again I get login screen and everything is gone, the system seems to shutdown instead of hybernating. I use this to hibernate:
sudo pm-hibernate
My kernel version is 3.2.0-36-generic according to uname -r and I have plenty free swap when I try to hibernate (almost all my 8Gb).
Thank you very much for your help.