It's possible the apps you are using aren't compatible with iOS 7.
Personally I see a springboard crash every few days, with random app crashes in between on my iPhone 5S. My iPad 2 doesn't have any springboard crashes, but most apps get killed because of low memory.
My guess is one of the iPhone only apps is triggering this. I'd bet on weather since it runs in the background periodically.
It could be Weather. I may try turning it off in background refresh. I know it's not necessarily 3rd party apps because I had to endure the 48 hour test of running the phone with strictly native software and apps only and it still happened. Buggiest phone I've ever had. This will teach me though. I long ago learned the lesson not to buy a new car the first year that model was out because all the bugs haven't been worked out. Guess the same thing applies to phones. This is my first iPhone that I've gotten right at release--my previous ones were all delayed (got the 3G when the 3GS came out and the 4S when the 5 came out) so all bugs were pretty much worked out. But I just find it weird that some people are having no problems or only occasional (and to me, having an occasional app crash that takes you back to the home screen is just a minor annoyance that can be dealt with) minor issues while I keep having major system crashes no matter if I'm running it clean or running it with apps. I'm really beginning to wonder if it's me....
But my appointment is in a little over an hour. I'll get the new phone since that's what will please the 'geniuses' and I'll spend hours getting it set back up again (I'm going to have my 25 digit Wi-Fi passcode memorized at this point), and I'll keep my fingers crossed that it all works as it should....but I won't hold my breath.