Oh, I've had this one too (on an iPhone 4.) As far as I can tell, it's caused by switching cell towers on 3G/4G; you suddenly get apparently-great signal strength, but no service whatsoever.

I've found that entering and leaving Airplane mode fixes things, presumably because it causes the phone to notice which cell it's actually connected to now.

In order for this to be effective in the app you have open, though, you have to stay in Airplane mode long enough to time out whatever TCP connections the app is attempting. Tower-switching is a purely physical/routing-layer effect, so you still have the same IP, and TCP connections stay established. If you trigger the carrier renegotiation without staying in Airplane mode for long enough to drop your DHCP lease (usually about three seconds, I think?), the phone will go on waiting for a reply to a TCP SYN that never actually hit the tower, so you'll find, like you said, that you may have to kill-and-restart the running app to get it to work.

I'm fairly sure I've had it on wifi, though I'll have to pay better attention to the circumstances and see if I can confirm that.

Doubt many people will read this, but an update:

Dropping internet connection does occur on WiFi and is not fixed in the iOS 7.1 update.

iTunes Music has been horrible for this. I'll stream some music, then hit pause and walk away for a while.

When I come back and hit play, most times it'll play out the buffer and then decide it has no internet connection, fixable only with a kill-relaunch. And some times it'll reboot the device altogether.

Oh that reminds me of a couple more. Tons of lag and crashes in the iTunes Store app, and tons of lag in Podcasts, along with chunks of the episodes lists showing up white when I knew there were episodes that should be there.

I also had an issue where around half of my music got corrupted during a sync. It acted like everything was fine, then as soon as I tried to play a broken song it would just skip it. Which was particularly irritating because it made "previous song" via 3-clicks on the inline play/pause button stop working. I'd have dead spots in the middle of playlists that I couldn't rewind past because it would hit them and immediately jump to the next song.

Resyncing seemed to fix the audio, but ran out of space because it left behind 4 gigs of cruft in "Other" that the storage page in settings couldn't account for anywhere. I had to do a restore to get everything working again.

So all in all, my experience with iOS 7 haven't been great.

EDIT: One more complaint in the design department, even though I don't mind the iOS 7 style as a whole. The new Music app is very clearly designed for the iPhone 5 and up with taller screens. The scrubber has gotten a lot harder to use, and if you're not precise enough you'll end up setting the song's rating instead.

For what it's worth, I've had horrible performance in the iTunes App Store since my first day with my first iPhone. And the switch to the new store 'style', heavy on pop-ups instead of directing to distinct pages, has been worse than before.

I don't think I've ever had an acceptably performant session in the App Store, at any time of day, on any device, from any network. Even direct links to app pages from other web pages have always been slow.

So I'm not one who'd be able to notice new 'lag'. But I haven't had any crashes.

The "Other" space tends to be MMS messages. If you're not manually clearing conversations, those pictures, videos and sound files are just getting filed away forever. They really need a fix for that. If nothing else, offering a setting to automatically trim conversations to only the most recent X weeks.

Not MMS here, I'm using Google Voice, and any MMS messages to that number just vanish into the ether. Only a couple of people have my "real" phone number.

The Other space ballooning definitely coincided with the music syncing bug. Before that it was a barely noticeable sliver in the space meter.

As far as App Store, that one actually runs well for me. It's only the music store that sucks.

The app store is so painfully slow even over wifi. It used to be much better before the switch to the screenshot-heavy layout. I like the screenshots in theory, but it is so slow that it makes browsing apps feel like a huge chore.