Two things you should do:
1. Using a multimeter, set it to the DC Volt mode (if autoranging), or 20 DCV (if manual ranging)
On the main (24PIN) PSU connector (while it's still plugged in to the motherboard), stick the black probe inside one of the black wires, and the red inside the purple. It should provide a 5V output at all times (as long as the on/off switch of the PSU is set to on)
this is the standby power the motherboard receives from it.
If it gives significantly less voltage than 5V, then that would clearly be a bad indication.
2. Remove the power cord, disconnect the 24PIN connector from the motherboard, and jump start it (link between one of the blacks to the green)
You could simply use a wire that is stripped on both ends, or a metal paperclip (if using a metal paperclip, make sure the connector is placed under a non-conductive surface, such as a piece of cardboard), put the power cord back in, and see what happens.
To put it simple though, if it doesn't turn on, the PSU is bad and has to be replaced, but if it does turn on, then you got a few different possibilities:
A) Still could be a bad PSU (check the gray wire on the 24PIN), it should provide a 5V output while powered on (which is also the "all clear" signal for the motherboard)
B) Could be a bad power button on the case, to test this, simply switch between the power/reset wires on the motherboard.
This would enable you to use the reset button as the power button. If your case has no reset button, you could simply cut the wires of the power button a bit above the connector, strip em, and twist em together (to create a power on signal), if this would not make a difference, just put it together again and use electrical tape, simple enough.
C) Could be a short between the motherboard to the case, try removing the motherboard from the case and placing it on a piece of cardboard.
If the problem still remains, then the motherboard is dead.
Good luck!