Within the Hip Message System, messages you've received or sent can not be deleted; however, you can archive them, to remove them from your main Inbox and Sent Inbox.

Archiving Messages in Bulk from the Inbox

From within the Members Area > Messages page:

HipStamp: https://www.hipstamp.com/members/messaging/browse/filter/received
HipComic: https://www.hipcomic.com/members/messaging/browse/filter/received
HipPostcard: https://www.hippostcard.com/members/messaging/browse/filter/received

You can check off the checkbox to the left of any message (or select the top-most checkbox to check off all) and then scroll to the bottom of the page, and click on the grey "ARCHIVE" button to archive your messages.

Archiving an Individual Message

Once you have opened a specific message, from the view/response page, you can archive an individual message by clicking on the grey "ARCHIVE" button which will appear to the far-right of the heading labeled "Message Board".