iOS & Android:
To delete personal messages in the app, just press and hold the message you would like to delete. When prompted with a pop-up confirmation, tap delete. You can delete a thread or messages, or just a single message.
To delete messages while signed in on a web browser you can click on the 3 dots on the right-hand side of each message thread. You can only delete message threads while signed into the web app - use the mobile app if you need to delete individual messages.