Hibernate appeared, but fruitless!
Apparently this forum has no "edit" feature, so sorry for this double post.
After setting from S3 to S1 in BIOS and restarting, the "Hibernate" button indeed appeared in my Start menu. It did seem to enter some mode, as my screen went to standby, but the CPU fan was still active. I definitely want a mode where it wont be, because I also want to have some silence from time to time.
Besides, I was unable to wake my PC up again, like I feared. Keyboard, Mouse, Power button, nothing worked, I had to hard-reset.
What's that about driver-hunting? Obviously I got the newest ATI drivers from their homepage, but I still have this problem. Are there any other drivers of note that can solve this?