It seems the problem arises when using the iWork apps: Pages, Numbers and Keynote. Reports say the problem happens when the Home button gets a double tap to exit an iWork app. The result, in some cases, is a blue screen and then a reboot of the phone.
I could not duplicate the problem with any of the iWork apps on my 5s, but it's clear some users are experiencing this frustrating bug.
Like any new release, iOS 7 hasn't been trouble-free. Apple is already working on a nasty iMessage bug that prevents texts using the Apple servers from being delivered. It's likely iOS 7 will see an update with some bug fixes shortly. Until then, Apple has inadvertently given you a bit of Windows nostalgia on your iPhone 5s.