Like you my iPhone 5s, iOS 7.1 just makes me want to throw it out of the window because of frustration on the touchscreen. It all happend after I updated to iOS 7.1.
I found this post from Ozboyblu (https://discussions.apple.com/thread/5987552?start=30&tstart=0) which ended up having Apple to replace the unit.
Unfortunately for me, I bought the phone when I visited the US and now that I am not in the US, I'm stuck with the crappy iPhone that used to work before the recent update.
If somebody from Apple is reading this, please work on patching this issue ASAP.