In my humble OPINION:
Firstly, if you ask someone with a particular service, they will usually tell you the service that they use - or they wouldn't use it. There are exceptions - usually someone who has mistakenly switched and wished they hadn't. One can usually get a strong signal from the brand name carriers in large metro areas and on main highways. This makes it sometimes hard to determine coverage. If you put any faith in Consumer Reports (in many cases I think their ratings have out-of-date models and information), you would undoubtedly go with Verizon. Last time I checked CR, they had the highest service satisfaction in I think all continental US states. The second rated (overall) was AT&T, BUT in several states they were THIRD to Alltel or T-Mobile! In my area - lower midwest - I have used AT&T, but mostly Verizon. I have dropped less than a dozen calls in the last two years with VZ, and I even travel into some rural areas. I have friends that continue to drop calls with AT&T daily. That being said, if you travel overseas - AT&T may be better.
I almost switched back to AT&T due to PDA data plans being required with Verizon, planning to deal with the frustration of additional dropped calls. We have a four line family plan and AT&T wanted a $35 activation charge per line!! I don't recall ever being charged an activation fee with Verizon and have even switched several phones.
In summary, go with Verizzon unless you have a family that goes unchecked on minute useage. With AT&T, the rollover minutes can be a great benefit to cover this. The plan costs are competetive for one vs the other. If you manage/monitor your minutes- and you can do this very nicely (per line even) by registering online with Verizon - the call quality, Customer Service by phone and online support are superior with Verizon. I also have a strong preference for LG phones, especially on Verizon.