3 options:
Power switch: Since the ATX power switch is a simple momentary contact push button switch, they can fail and complete the circuit on their own.
Power Supply: If the power supply control circuit starts to fail, it can switch on the system by itself.
Remember, an ATX supply is on ALL the time. It switches to full on when instructed to do so by the motherboard.
Motherboard itself: Either too much current draw on the 5 volt standby line, a failure in the power control circuit, or a wake-on event is happening.
Start by entering BIOS setup and disabling ALL wake-on events in power management.
After power up, disconnect the cable for the power switch. then shut down and see if the system stays off.
You can use an ohm-meter to check how sensitive the power switch is to operate, or if it has a resistance of less than 1 Megohm when not pressed.
That then leave the power supply or the motherboard. Swap in a different power supply and see if that fixes things.
Chas