I was recently asked about saving voice mail messages that are on an iPhone. Specifically, someone had received an especially cute message from their son, and she wanted to keep it for posterity. That's a cute idea, so I looked around for a solution, which I found, and I'll explain here.
We'll pull this off with a piece of software called iExplorer. iExplorer can access the voicemail files that are on your iPhone, and export them to your Mac. There's a free demo version you can use with some restrictions, or unlock all functionality for $34.99 (note that what I explain here will work with the free demo version). Here's how to save that voice mail message and really, it couldn't be easier.
First, install and launch iExplorer. Then connect your iPhone to your computer with its cable. The app will recognize it right away.
Now click Voice Memos and you're brought to a list of all the memos that are currently on your iPhone. Click any one to get information like file size and date, or to listen to a preview, just to make sure you've found the right file. Finally, click "Transfer from Device" and select your preferred destination. That's it! Now you've got a .m4a file to cherish forever.