I have a friend who is an Apple genius. I asked him to take a look at my phone, and he told me what the issue is. The issue is with springboard and that's an issue that is common with people who upgrade from an iPhone 4 to a 5s.
I have an iPad retina display first-generation an iPhone 5s which I just upgraded from an iPhone 4 and I have this issue with both my iPhone my iPad and also my wife's iPhone and my wife's iPad are affected (My wife has an iPad mini).
Springboard is part of the operation system and what causes this is a corruption in springboard. From what he told me it cannot be solved just by synching or even doing a clean reinstall.
Yes it's a software issue but if there is a corrupt file in one of your apps then you are transferring that corrupt file every time you synch. That is why this problem apparently is very difficult to solve. He says that he has seen other Apple geniuses just give people new phones because they can't deal with the issue at that moment and the people come back again in two weeks with the same problem.
Basically the solution is the following:
You need to do at clean reinstall of the software as if it was a new phone, but do not do this from the software that you have in your computer. You need to re-download the OS. You also need to reinstall every app in your phone but do not do it through a synch. You need to download every app manually. If you synch your old apps, the problem will not go away. He also said not to download the apps directly through the phone, since sometimes when you do it that way files can become corrupt in the process.
He told me it's a pain in the ass but it's the only way to deal with it, and obviously most people don't want to go through that.
He also told me that it's probably not going to be something Apple fixes with an upgrade anytime soon. He said the issue is not that widespread or that common so Apple has not dealt with it through an upgrade.
I myself have not tried this. It seems like something that would require a great deal of time and I have not had time to do it. If somebody else has tried it please post the results.
I hope this helps.