Gmail fix:
I tried all the fixes here for an identical issue with my gmail account, but kept being left with 26,000+ allegedly unread emails on the iphone 5 after upgrading....
Now, it seems to be OK after the following routine, done on my PC (NB not sure how applicable this is to other email accounts)
1. (very important) - click on All Mail from the gmail menu on the left hand side of the screen (if you can't see this, it'll be revealed be selecting 'More' from the menu list). (It would appear that the emails that the iphone is referring to are not, in fact, just the ones in the inbox..... )
2. Click the 'down' arrow on the right hand end of the gmail search box (it's just by the blue button with the magnifying glass)
3. In the 'Doesn't Have' field in the Search menu, enter some gobbledygook - like "asgjihg567vfd" - something that you are certain will not be in any email. That means that all of the mail, including long-term deleted mail (not just the stuff that appears to figure in your bin), will get selected
4. In the left hand of the three grey boxes above the emails (the one with the square on it), select 'All'
5. You will see on the bar above the emails that a message comes up saying - "All 20 conversations on this page are selected.Select all conversations that match this search". Click on this to select, literally, all your emails - past, present, binned, spam, etc.
6. From the right hand control box above the emails (the one saying 'More'), select 'Mark as read'. This will take a few moments, but will work through them all.
7. When you go into the gmail account on your phone, you should see the unread email count miraculously disappear when it syncs.
This has worked for me and I hope it stays this way! The trick is that you may have lots of unread emails that you binned ages ago - but these have been resurrected through the miracle that is iOS7.
FWIW, I very much regret upgrading to iOS7 and have written to Tim Cook telling him that it's going to lose him customers - not that I expect that he will see my mail, or answer, or that he cares.......