If you're certain it was the power supply that, umm, "exploded," plug in a different power supply and see if the motherboard comes back to life. Not much else you can do in this case, as test equipment for motherboards is not something a consumer is going to have access to.
Power supply testing equipment is available for reasonably inexpensive prices, but depending on the symptoms, you likely already know your PSU has gone kaput and won't gain much by testing it.
Low cost PSU tester:
http://tinyurl.com/ob5un96
While any damage is likely already done, if the original source of the problem was not the PSU but rather a faulty piece of equipment or connection somewhere else in the system, plugging in a new PSU may not yield the desired results. Generally, a good quality PSU will shut down if something shorts out, but that may not be the case for cut-rate models. Be prepared to remove power to the system on short notice when first powered back up.