You can delete your posts. You can't delete the post if it is the foundation of a thread. If your post qualifies, you should see the delete option available to the right of the edit button.
Edited to add: The perk should be used with caution. If the staff believes a member is abusing the perk, one may see a message from the staff in his or her inbox.
That doesn't show a post that has been made. It shows a post in the process of being created. And if it was the start of a thread that was posted, the mods would have to delete it.