Same problem here. It usually occurs from the lock screen or when using TouchID to unlock. It's been growing in frequency and I submitted a Bug Report with crash logs to Apple.
A restore does not fix it. This is a bigger problem under the 64 bit architecture of the 5s. I was running 7.0 on my 5 with no such problems.
This has been happening to me too. Yesterday it happened when I asked Siri to play a station on iTunes Radio. Today it happened when I tried to start playing an album I had just purchased from iTunes. What gives, Apple?! It's happened probably 6 other times, but I don't remember what the circumstances were. I have a 64GB 5s AT&T model.
Same problem here , no matter I have reset and restore I am also having this problem . At least once a day , my family also having the same issue while they using iPhone 5s
CrashReporter Key: a887fc8f6e6130612eeed8e7a2aafd67e3049eec
Hardware Model: iPhone6,2
Process: SpringBoard 
Version: 50 (1.0)
Code Type: ARM-64 (Native)
Parent Process: launchd 
Date/Time: 2013-12-19 00:29:35.991 +0800
OS Version: iOS 7.0.4 (11B554a)
Report Version: 104
Exception Type: EXC_BAD_ACCESS (SIGSEGV)
Exception Subtype: KERN_INVALID_ADDRESS at 0x00000000fd3bbeb8
Triggered by Thread: 0
I'm sorry but nothing can fix the problem definitively (a restore, reduce motion on, or any other option).
It is an iOS bug! I' experiencing the same problem on my iPad air, and if you want to know something more you can read this discussion: https://discussions.apple.com/message/24225868#24225868
There are a lot of people experiencing this issue, with all device (old iPad and iPhone too), especially with 64 bit version.
Some users have changed their device 5 times but the problem is still there.
However, it seems that Apple is working on the problem. it is hoped this issue will be fixed in iOS 7.1 that is currently in beta testing with developers. Yours story is the same most of us have had.
It is unacceptable.....and very frustrating...but the good notice is that it isn't an hardware problem.
It is harmless for hardware and data. Your device will just crash or maybe restart the springboard, losing nothing.
It seems that ios7 currently does not release the ram in an efficient way, like previous iOS did... Moreover there are some problems with some native apps. This is the idea I made reading other discussions on this forum and surfing the web. I don't know why these issues are "most popular" with 64bit devices....maybe these devices need more efficiency? Maybe the combination of different bugs?
However, I read that some users have spoken with Apple senior advisor who have captured error logs file. So Apple is working on the problem.....I hope.. Meanwhile, it seems that a restore through ITunes mitigates the problems...
At any rate, it is not what I expected from Apple and from a 700 euro or dollars devices... It is a shame: no one word officially from Apple!!!!
We can only report the problem to genius or senior advisor, or return the device and get another iPhone with the same issues, and you will have a new return date.
Apple messed up big time. I'm very disappointed...