In TextSecure, you can choose to backup your SMS to the SDCard.
It says "Success" but there is no evidence where it does store it on the SDCard.
I cannot find it.
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Like Chahk said, the folder is
/sdcard/TextSecureExport. I've had trouble accessing it (even after moving it) over the camera-centric USB interfaces.
I ended up compressing the file to a Zip archive using ES File Explorer and then using Google Drive to browse and upload the file. This also makes it easy to pull back down to a different device or after wiping memory.
The folder is located just in the base dir of your SDcard:
Since Textsecure is now called Signal, there is no extra folder. It is located directly in your SDcard home folder:
Note: sometimes the phone doesen't show the folder on your computer, then you could use a filemanager on your phone (like Ghost Commander) to move the backupped files into another folder that is visible on your computer.
My Android Signal app version 4.10.12 (date 2017/10/01) saves media of conversations in Device storage > Pictures.