An Update:
I am on Verizon (I'm updated with the latest Verizon carrier update), I have an iPhone 5 (64 GB) and my company uses MS Exchange 2010. I was coming from a 4S on AT&T that worked perfectly. On Friday (9/21) when I got my phone I synched to my iCloud Backup and my phone was set up a couple of hours later, minus all my music and non-photo stream pictures. Saturday/Sunday I started noticing that I wasn’t getting any notifications.
Monday morning at work I wasn’t getting anything and that's when I stumbled upon this page. I too was suffering with apsd crashes(10-15 per day since 9/23), some low memory crashes, etc. and no push on Exchange and then no push notifications all together (nothing from AP Mobile, Songpop, FB, etc.). My iMessages were sending green (as a text, not iMessage) and a reset/reboot/ or Airplane mode toggle would fix my phone for 5-30 minutes. I would also get all my push alerts at once after the reset. It has been very frustrating.
I called Apple and spoke with several tier 1 and tier 2 individuals. They called me back multiple times and followed up with emails from their engineers.
My last call was this past Friday (1 week of non push frustrations). After talking with a Tier 2 individual, I went into Settings, then Messages and limited my Send & Receive Addresses to 2 and did the same thing to my MacBook Pro and MacBook Air (as I thought the conflicts here might be causing some of the crashes). I did the same thing with FaceTime on my phone, Macbook pro and Macbook Air. I also went into Settings, iCloud, Account, Mail and turned OFF my @cloud.com email (I do believe this was an important step).
At this time (after those setting were saved to iCloud) I went to Settings, General, Reset and "Erase all Content and Settings" and started my phone fresh. I connected to iCloud (with the new limited number of iCloud , Messages and FaceTime Send & Receive Addresses). My phone worked well and was very zippy with iMessages and FaceTime.
About 3 hours later I connected 1 MS Exchange account (MS Exchange 2010) to my iPhone and I downloaded the GMail App and used that for my Gmail accounts, since It has push notifications.
So far since Saturday, I have had push working well without any issues. I have forwarded all my GMail and other accounts to iCloud and all is working well (since this was working so well I deleted the GMail App).
So, I currently have 1 Exchange and my iCould email push synching to my iPhone 5 and all is working well.
I don't know if this will work for everyone, but if you follow all my steps I hope it works for you.