If you mean private messages, there is an option at the top to leave it. I'm not sure it will fully delete it but it won't be in your messages anymore.
And I don't believe you can delete notifications, either. You can open the notification alert and it will mark them as read. Then they will stop displaying, but can still be found on the notification page.
Your only option is to turn off notifications. When you open the notification box there is an option that says 'Preferences' that takes you there. This gives you the option to opt out of different sorts of notifications.