If my opponent has a permanent which I take control of, and it is then destroyed, does it enter my graveyard or theirs?
Theirs.
The long version:
404.1. A player's graveyard is his or her discard pile. Any object that's countered, discarded, destroyed, or sacrificed is put on top of its owner's graveyard, as is any instant or sorcery spell that's finished resolving. Each player's graveyard starts out empty.
Ok thanks for clearing that up