You can check the power supply without the motherboard.
Disconnect the power connector from the motherboard. This should be the 20 pin or 24 pin connector. Also disconnect the 4-pin connector also, if there.
Use this
Info here on pinout to short Pin 16 to Pin 17. The power supply should turn on if its working, also the power supply fan should be spinning.
If the power supply is working, then you have a connection problem on the motherboard or a motherboard problem. Check the power button, make sure its on the correct front panel header pins of the motherboard. (i.e. where the HDD LED, speaker, reset, etc.) plug into. Reseat all components to ensure a snug fit.