I wonder if it is possible to power up the OPIs with the power supplies from classical PCs, like ATX ? What would be the wiring to switch on the power and bring the 5 V to the SBC ?

you would need something like this:

http://www.ebay.com/itm/321914510023

for the power connection.

To run the opi, i connected ground (black) and +5V SB (Pin9, purple) to this connector.
My ATX supply delivers 2.5 A on standby, without even being switched on.
This way the opi runs silently.

via software, using a GPIO pin and a Transistor you can pull down PS_ON (Pin 14 or 16, green)
to start the power supply and power LEDs or motors or such.

This is a great knowledge. Thanks.  

You can  connect the +5V SB pin to pin 4 on OPi gpio header, and ground to pin 6.(this work on my setup)