I use a programme on my PC called Touch Copy. Their website is
http://www.wideanglesoftware.com/touchcopy/index.php
It's reasonably accessible but a bit slow. The SMS or iMessages are stored as HTML files, the filename being the number or the person's name you were messaging with. This filename can of course be changed once it's on your PC. I change it to the date and put it in a folder with all other conversations with that person.
The programme will allow you to back up pretty much anything from your phone so has a wide variety of uses although I've mainly used it for iMessages and SMS.
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Backing Up