I have an iPhone 4S with iOS 5.x: Two nights ago, I lost the 3G network, with the message "Searching. . . ." replacing the usual Verizon 3G notice. (WiFi is fine, however). The network came back for a while later and has been dropping and returning intermittently. I reset the phone, but that did not help. I went to a Verizon center yesterday morning and was told that there had been some network "issues" because of the transition to iOS6. The associate "reset my network", which she said was different from my reset. The network returned for a short period, but was "Searching" for most of the day.
I went to an Apple Genius Bar today, where they ran diagnostics and told me that the phone was OK. That person also did some re-sets and advised me that some applications may have been remaining in memory, and conflicting with the operation of the phone, but that any issues with Verizon's network were beyond them. I've had the phone for 11 months and there have been no connectivity problems before 2 days ago. Again, WiFi is working fine.