Public transportation in the Vatican

Encompassing less than a square km, the Vaticano has no transport of it's own, save the heliport which is exclusive to the Papal travels!

To your question, 'Do you use them?', unfortunately, I couldn't! : )

On a serious note, I didnt see any local transportation in Vatican, other than the tour buses that come in from Rome.
Vatican can be crossed within 20 minutes on foot, so a local transport does not seem like a necessity. Though I have heard of a train used for special occasions.