There are several ways to do so.
You can sign up for Google Voice to handle your voicemail, and they will send you an audio file with the message if you choose your settings to do so.
You can use an audio cable from your phones audio port in plug it into your computer to record your voicemail to your computer.
Verizon gives you instructions on how to have a 3rd party record your message and send you the file for a fee.
I am sure there are other methods.
That's a great question! Rschnoor provided some great options, and there's one more I want to point out. Visual voicemail allows you to forward messages to an email or save to an SD card. Here's some more info on the different types of voicemail and what they do http://vz.to/YH66Ge.
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Keep in mind that the Verizon Visual Voicemail will add an additional $3 to your bill every month. Just another way to inch your bill ever higher.
You can record and save all of your voicemails as mp3 files using Voicemails Forever (www.voicemailsforever.com)
Btw, the link is here: Voicemails Forever http://www.voicemailsforever.com . You'll get 60 minutes of voicemail recording time for $19.99 and then mp3 files of your voicemails. Of course, you can also sign up for Youmail or another service afterwards for FREE to download FUTURE voicemails here as well.
All the best
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