Is It Safe To Test A Pc With A Possiblly Burnt PSU
No, don't plug it in. The "bang" was a component getting destroyed. Hopefully a fuse, but likely something else.

The danger (other than getting shocked or starting a fire) is that the PSU's output voltages may no longer be correct. This could destroy your motherboard (bad) or hard drives (worse, in my opinion).

Of course, this may have already happened...

If you have data on your hard drive that you want to recover, I recommend buying a "usb hard drive bay" (or "usb hard drive docking station") (~$15USD) and using another computer to access the data.  These bays provide a level of isolation between a possibly-bad hard drive and the computer.

You could simply connect the hard drive directly to a different computer, but this risks damaging the new computer's power supply if the hard drive has developed a short circuit...

After you save your data (or not, if you don't want to), you can simply replace the power supply and see what happens :)