You can leave a DM thread which removes it from your DMs but there's no option to actually delete it. Kind of a legacy of when they were "Conversations". Not sure what happens if there's only 1 person left in a DM thread. I guess it would stay in that one inbox until the last person leaves it?
Description: Allows searching all conversations. This add-on is designed for admins to search and view all conversations. Note: if you're running XF 2.3, you will want to use Direct message search plus add-on...