BackupTrans iPhone SMS Backup and Restore is an iOS SMS transfer tool. It can be used to back up individual messages from your Messages app or all of them at the same time. When you connect your iPhone to your Mac with its USB cable, this application will detect it and display the name of your iPhone on the left pane of the main window. When you click on your device, the right pane will be populated with a list of your SMS messages, which you can sort by date, contact name or phone number. You can check a box next to the messages that you want to backup. Below the message list is another view which shows all your messages in a way that is similar to what you use on the Messages app. The messages that you sent are shown on the right column and the ones received, on the left one. You can select any text here and copy it to the clipboard. That way, you can copy your entire conversation history to a single TXT file for safekeeping.
If you click on the little arrow next to your device name, you can narrow down the messages shown by contact. A list of all your contacts who have messaged you will be displayed under your device on the left pane, and when you click on a name, the messages will appear on the right.