Debbie
Old or new, a question can't be self-deleted if it's been answered because the option to delete is removed from the edit button the moment the first answer comes in. Once this happens, the only way it can be deleted is if it's reported as a violation by other users. What you can do though is use the "maker anonymous" feature found in your edit button to remove your username and avatar from it, this is especially useful if your question is embarrassing.
Anonymous
If it has one or more answers, you can't.
You can make it anonymous, though.
If you violated your own privacy, you can report it, and it may get deleted.
About every 5th or 6th question in the Yahoo Answers category is "How do I delete a question?". Sometimes you can find the answer to your question without having to ask.
Son of T3
If the question has answers, the only real way to remove them is as violations. You will need to get friends and network to report. The easiest thing to do is go to the question and change the asker to anonymous. You will then be the only one who knows it is you. The staff in the past would remove one or two, but no longer.
jeligula
Once a best answer has been chosen, the question is archived and cannot be deleted. It then becomes the property of Yahoo!