You can delete a comment as long as you have DELETE permission on it, and RELATE permission on the discussion. These are your permissions when you author a comment.

When you delete a comment, any replies to it (and replies to them) are also deleted, regardless of your permissions over them. If you have DELETE permission on an object, you may delete all comments in its discussion, even if you lack permission to edit those same comments.

While you cannot explicitly delete a discussion, deleting all of its parents effectively deletes the discussion as well.