Apple - Can you share or save iPhone voicemail messages?

I believe there are applications that can extract the voice messages from your phone but they cost money and required the phone be plugged in to the computer.

The simplest way is to connect your phone to your computer with a 3.5mm audio cable. Use a simple audio recording app, such as Audacity, click record and play the voice message.

I used a third party app called TouchCopy to save my iPhone voicemail messages to my computer.

TouchCopy allows you to play and copy Voice Mail messages you have received on your iPhone. To view all Voice Mails stored on your iPhone, click the Voice Mail button in TouchCopy.

TouchCopy will copy messages from your iPhone if your iPhone supports Visual Voice Mail. These will be voice mails that are provided as part of the Visual Voice Mail data plan from your mobile service provider.

^{Source: Copying iPhone Voice Mails with TouchCopy}

Note that the voice mails actually need to be stored on your iPhone rather than by the network operator for it to work.

I have a Google Voice number that I just forward important voicemails to.

To do this in a typical voicemail system, you'd press 6 for more options, then 2 to forward it.

You can keep them in your Google account pretty much as long as you want, or you can simply download them at will.

I hope this is helpful.