I thought it might be a better idea to create a separate thread for this problem.

Do you guys know what might cause a PC to remain turned on after shutting down Windows? I have all drivers installed and the Device Manager doesn't show any problems. Suspending the PC (S3) works fine (except that it takes half a minute for the PC to wake up) but if I tell Windows to shut down, the fans keep spinning and the power LED is on.

Start the device manager (Win + Pause key -> Gerätemanager) and check that you have something with ACPI listed under Computer. If it says Standard PC there, you didn't set ACPI/APIC in the BIOS (which you should).

Will do, thanks. However, isn't ACPI required for S3 (Suspend to RAM) and S4 (Suspend to Disk / Hibernate) to run?

Problem solved! The reason why the PC remained on was the Gigabyte SATA RAID controller which had the wrong drivers installed. During the installation I selected IDE operation mode, but the BIOS setting was AHCI. After replacing the driver everything works fine.

I was just about to say after reading the OP that of Vista's many quirks, drivers hanging at sleep or shutdown are fairly common. I had a similar issue with one of the Dell system utilities on my XPS keeping the machine from shutting down properly - I solved it by not letting it run in the background all the time, but opening it only when I needed it, and closing it when I was done.

I'm glad you solved your problem.