Ok, thanks. I confirm there is not much going on which seems to be related.
The fans almost never operate on my computer, so hard to say. It seems the system is working in the background. For example, if Facebook is launched in Firefox and a message arrives, I can hear the sound of it. Based on Techrepublic description, it looks like a normal behavior: "At S0 the computer is in a Working power state and the system is fully operational. Even so, certain devices that are not currently being used can enter into a low power state to save power." That is why it takes milliseconds to go out of this mode. I am still confused by the result of the command powercfg -lastwake. Maybe S0 is not considered as a sleep mode by this command?
I am with you, and I would prefer S3. I am going to switch to hibernate or just turn off the computer. Maybe I am too old, but I don't see the advantage for me to use S0.