Most voice mail messages have served their purpose after you listen to them once, so leaving them on your iPhone takes up space for no reason. Deleting voice mails is a two-step process -- first, remove them from the main list of voice mails and then delete them permanently.
Open your voice mail box and tap "Edit."
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Tap each voice mail you want to delete to select it. This is necessary even if you want to delete all of them, since there is no way to select all voice mails with one tap. Tap "Delete." This removes the selected voice mails from the voice mail box and places them in a separate folder.
Tap "Deleted Messages" to open the folder containing removed voice mails. If you left some voice mails in the main box, you may have to scroll down for this option to become visible.
Tap "Clear All."
Tap "Clear All" to confirm and delete all removed voice mails permanently.
To undelete a message and send it back to the main voice mail box, tap the message, and then tap "Undelete." If you've already removed the message from the "Deleted Items" folder, the message is not retrievable.
To customize your voice mail greeting, tap "Greeting" on the main voice mail screen, select "Custom," tap "Record," record your message and then tap "Save."
Information in this article applies to iPhones running iOS 8.1. The process may vary slightly or significantly if your iPhone is running a different version of iOS.