Q: IOS 10 blue screen

Get help with the screen on your iPhone, iPad, or iPod touch - Apple Support

If you see a solid color screen (black, white, or blue)

1. Plug in your device to a power source. If you see the low-battery screen when you plug in, let your device charge for at least 10 minutes (20 minutes for iPad) before using it.
2. Force restart your device while it's plugged in.
3. If this doesn't work, restore your device.
4. If you still see the issue after you restore, contact Apple Support.

Try the second suggestion.

I have this problem on iPhone 5S too after installing iOS 10. While doing force restart i have blue screen and then my phone boots like it should and works without any problems. Previous versions of iOS didn't have this bug.

Me too...

Did you try restoring the device?

In that case, I think the next step left is to contact Apple like it states in the 4th suggestion. There is nothing more that you can do on your own.

Contact - Official Apple Support

I have the same issue after ios 10 update on my iPhone 5s.   It's so super annoying as well as alarming, seeing the bright blue screen for a couple of seconds  before the system reboots.   Tried  complete phone reset and re-installed the phone a couple of times using iTunes,  and a couple of times using ICloud, (Total of 4 times!) wasting almost 2 days!!!  The system otherwise behaves as it should (I think a little faster too) ...so far, at least!!!

Hope Apple comes up with a resolution in the next update!!!

So the best thing to do is what Demo was saying earlier, which was restoring your devices. If none of that works. Like Demo mentioned earlier the best bet is to reach out to Apple Support and see if there is more that can be done.