I've had the same problem beginning Wednesday, January 8,2013. The first S4mini was deemed DOA and I got a new one Saturday, 1/11/2013. All was well until I turned it on this morning! It came on in SAFE MODE-ground hog day! The tech support cleared it up like this:1. Power it OFF
2. Press at the same time: UP VOLUME/HOME KEY/and POWER BUTTON.
3.Watch the top left corner of the screen for the tiny bright blue message about
REBOOTING when this appears, LET GO OF THE THREE BUTTONS.
4. Use the Volume button to scroll to the 4th item(some kind of reboot-I can't
remember the exact title)
5. Use the POWER button to SELECT it.
You'll see the little android robot falling down, getting up with a spinning tummy and at the end of his act the phone will be off.
Power it back on and hopefully you'll be off of SAFE MODE.
The tech told me to definitely call back(Verizon) if it comes on in SAFE MODE again tomorrow. I always power down at night and charge up.
I had two phones in less than 7 days do this-something is up.