Q: iPhone 5 randomly shutoff and won't turn back on!?

See Here for  >  Frozen or unresponsive iPhone

Try this First... No Data will be Lost..

Press and Hold the Sleep/Wake Button and the Home Button at the Same Time...

Wait for the Apple logo to Appear...

Usually takes about 15 - 20 Seconds... ( But can take Longer...)

Release the Buttons..

Try it again...

If no joy... Try a Restore...

1: Connect the device to Your computer and open iTunes.

2: If the device appears in iTunes, select and click Restore on the Summary pane.

Restoring  >  http://support.apple.com/kb/HT1414

3: If the device doesn't appear in iTunes, try using the Steps in this article to force the device into Recovery Mode.

More Info in the link from my first post >  Frozen or unresponsive iPhone

See Step 3 above.

Make sure you have the Latest Version of iTunes (v11.1.5) Installed on your computer

• iTunes free download from www.itunes.com/download

Any Error Messages or Alerts...? Are you using a PC or a Mac...?

The Basic Troubleshooting Steps are:

Restart... Reset... Restore...

As you have tried all these Steps and you still have issues...

Then a Visit to an Apple Store or AASP (Authorized Apple Service Provider) is the Next Step... Be sure to make an appointment first...

Thanks!  Worked for me