Open the message you wish to delete. At the bottom under the quick reply box, tick the Delete this Message check. Then click the Delete this Message button.
Note you may also need to delete PMs that you have sent, not just received.
Oh yeah, it's several steps. At the bottom you have to click the thing that that says "Selected messages" then click "delete" then click "proceed." Carpal tunnel, here we come!
It took me a little while to figure out How to do it on the mobile version too. I had no problem ticking off the boxes to the left of each thread I wanted to delete, but then I was stuck trying to figure out how to delete it from there. Well I found the 'delete' button and it appears as an 'X' just to the right of the 'compose' button.