To delete text messages on the iPhone 6, tap Messages on the home screen. Delete individual messages by tapping on the message thread, holding the preferred message, tapping more, checking the preferred message and then tapping the trash icon.
To delete voice messages from the iPhone 6, tap the bottom of your voicemail list and tap Deleted Messages. Once the messages have been pulled up, delete them by swiping or tapping them individually, or by tapping the Clear All button on the top right of the screen. The screen then prompts the user to either cancel the operation, or to proceed with a red Clear All button to delete all voice messages.