I've had endless problems with the the AppStore app since upgrading to iOS 7.0 (and 7.0.2 thereafter) on my iPad 3.
The AppStore will often open with a white screen, with no icons or icon place holders on. The screen can stay like this for 30+ minutes. When the AppStore does eventually load, searches result in "Unable to Connect" errors more often than not. Resubmitting the search often results in the appstore freezing (although the busy icon continues to animate). Downloads often start, but never download, leaving a grey place holder icon on the desktop.
I've tried:-
Closing all running apps and restarting my iPad. This has no effect.
Resetting all settings and restarting improves matters, but only for a few searches.
Logging out of my apple id, restarting my iPad 3 (by holding home+sleep/wake until the apple logo appears) and then opening the AppStore. The AppStore opens fine and I'm able to search without any problems. Strangely enough, previous uninstalled app purchases still show up as cloud icons, which suggests the iOS 7 AppStore caches data locally. If I download an app (at which point I'm prompted to sign in) the app downloads fine. However, the problems return on the very next search.
Restoring my iPad's firmware with/without restore from backup. The AppStore works again fine for an hour or so after signing into my apple id, then the problems return.
I strongly suspect iOS 7 caches app purchases. The evidence for a cache can be seen by signing out of your apple id, restarting you iPad and searching the appstore for an app you own: The download icons are shown as if you were signed in, right up until you try to download something, at which point you rightly get get asked for your apple id. Also, from the moment you sign into your apple id, "other data" grows continuously at a rate of 0.1Mb every 2 minutes.
I also suspect that the more apps (free or otherwise) associated with your apple id, then the larger the cache grows. At some point it probably gets too large for the iPads memory (not SSD space), at which point the app store becomes slow or unresponsive.
I had no problems using the iOS 6 app store (or indeed the iOS 5 app store for that matter). The problems started the moment I installed iOS 7. The problem isn't connection related either: iTunes connects to the appstore fine, as does my iPad right up until I login my apple id.
I've reported this to feedback and contacted support to no avail.
I sincerely hope Apple have a rethink about caching. The AppStore is their bread and butter. Anything that impairs access to it costs them money (I can't purchase an app if I can't open the AppStore!).
I wouldn't mind so much if I could revert to 6.1.3 while all of these problems are worked out (for 7.1?), but as it is, I'm now stuck with iOS 7 whether I like it or not, and whether it works or not.