My phone fell and the screen blacked out. The back button light still worked and I would see the indicator light above the screen blink from time to time indicating the phone was on, just no picture. I tried the soft restart (holding the power button and volume down together). Phone would restart and turn back on but no display. I tried charging the phone, and that did not make the display appear. I tried smacking and knocking the edge of the phone. I tried the squeezing the sides the phone with no luck.
What I did that worked, it may be out of luck, was squeeze the bottom of the display screen where the home screen & task manager button would have been located (bottom, center of the screen) and it would light up. But the display only lit up when some pressure was applied to that area. Attempt at your own risk.
That was in enough time for me to have someone else swipe on the screen and back-up photos/videos, while I maintained pressure to the phone. Also, I turned on Pandora to keep it playing to drain my battery as much as possible per Teo's 2/27/2015 post above.
What happened next made my New Year! After a while, of doing that multiple times, the display would not turn back on with pressure, however I knew the phone was still on as Pandora was still playing. So I booted up my computer and used the USB cord and connected it from the laptop to my phone, just for giggles...and the screen turned on as it began charging. When I unplugged the charger, the display remained lit and is currently working. The swipes and touch screen appear to be in working order.
Now, this may have been luck on my part and may be due to my specific issue; but I thought I'd share my experience in hope it will help someone else.