As a community moderator on Microsoft Windows Phone forums, I get to interact with people worldwide with various issues and one among them is the ability to backup SMS messages. Currently, Windows Phone does allow the backup of SMS messages, which is then stored on OneDrive and available for restore whenever disaster strikes. The problem with that implementation is that it does not allow the user to view the content of the backup. I very often see such requests on the forums, especially from users in the legal profession. I have some good news for those users.
Microsoft has quietly released a new app called contacts+message backup that lets you save your contacts in VCF format and messages in XML format to your storage card. No data is sent to Microsoft servers. However, it is very unfortunate that the app won't work on devices without a storage card. Most Windows Phones have a SD slot, but it does not necessarily mean they hold a storage card; Like my Lumia 1520 for example which has 32GB on-board storage and makes a SD card useless for my needs. SInce Microsoft budget devices are quite popular out there, it kind of make sense to go with the storage card.
Download contacts+message backup free in the Windows Phone store. After downloading, the app won't show in the app list, it will be in Settings.