After a hour of conversation with support, including Tier 2 Tech support (extremely empathetic young lady), it was determined that there is a problem with the current iOS (7.0.1) authenticating voice mail from the 5s. Unable to change voice mail password on phone, unable to receive voice messages on phone, no notifications of voice messages, unable to set password, or save greeting from phone. Only access to voice mail is through *86 feature.
I have seen numerous comments about this issue on other phones, but have seen next to nothing about 5s or iOS 7. Are there others having this issue?
If it is the phone, I need to know soon so as to get an exchange before hitting the road for work.
You might want to login to your Verizon account online, and try using the "reset voicemail password" link (not sure where it is exactly offhand), but that may work. Its worked for me in instances when I've swapped from an iPhone to an Android device, and then back to the iPhone (something to do with the voice mail system not recognizing it I guess), and has worked each time.