I'm on an iPhone 5s but have been experiencing massive problems just like this with iOS 9 and 9.1.
I actually started having problems with iOS 8 in May 2015. My first iPhone 5s (which was 6 months old) turned off one morning and could not be turned on by force starting, and was undetectable by iTunes.
The replacement was working fine with iOS 8, until 2 weeks ago when I updated to iOS 9. Since upgrading to iOS 9, my touch ID stopped working, and I would notice in the morning that the phone had restarted in the middle of the night, while on the charger. A few days later, I unplugged the phone from the charger at work, and it began cycling between the apple screen (about a minute at a time) and the blue screen of death (about 2 seconds) before turning off and beginning the cycle again. I attempted to hold down the home button and power button for up to a minute at a time, and nothing could restart it. It was 5 months old. To make matters worse, my 2010 Macbook Pro was getting repaired at the time, after crashing while trying to install El Capitan OS X, so I couldn't plug it into my computer to try and back it up to iTunes, either. I ended up having to take the 5s into the shop as well, and the phone was undetectable in iTunes. In a matter of a few days, Apple's updates had ruined both my Macbook and 5s completely.
Verizon provided me another replacement 5s (my third in a year) about a week ago, which has been now crashing 2-6 times a day. Periodically it turns off while I'm using it, other times it restarts on its on while charging, other times it just turns off and I have to hold down the home button and power button for a minute to get it to restart. This morning I had another flash of the bright blue screen of death, but it luckily came back on. Like many others, I'm thinking there may be something corrupt in my iTunes backup that the updates simply cannot handle, and am about to wipe my phone and restart. I took it into Apple 2 days ago, and they ran the diagnostics and the long list of crashes, and said there was nothing else wrong with the phone, just to update everything I can (the operating system as well as apps), as often as I can. The employee had me update to 9.1, saying it may fix it - it has crashed 7 times today, alone. I have little confidence in any of their updates fixing anything at this point. As a sidenote, my Macbook Pro STILL will not run El Capitan, and is still in the shop with Apple after 2 weeks. Diagnostics show "nothing wrong with it," a new hard drive has been installed, and yet it still crashes constantly since the update.
If anyone knows of a class action suit against Apple regarding all of this, please let me know. I'd love to join.