Firstly, have you ever turned this desktop on before? Is it just newly built? If it is, check your power button connector, maybe it's not hooked up right. If it's hooked up fine and still dosnt work, unhook it and short the two pins with a metal screwdriver and see if it turns on. If that dosnt work, try a different power strip or different power socket, or even a different power cord. If the fans spin up, it means your computer is getting power and it's not ur power supply.
If it still dosnt work or you have turned it on before on that power socket, triple check all your power connections check for damages. If you have a spare power supply, even an ATX works, just sit it outside of the shuttle and plug the large plug into the mobo and 12v line into the mobo and see what happens. Always check to see if that particular component is the problem if you buy. Scrounge up an old Power Supply, or borrow one from your friend. It's always better to double check instead of goiing out and wasting ur money.