It's #1 when referring to "the Underdark" in the Forgortten Realms or Greyhawk (ad possibly Eberron), because it's used as a nation\country name, like "Russia"...but if you're just referring to "the underdark", the caves and whatnot below your world, then it's probably #3, just like "the swaps" or "the forest" (as opposed to the "Great Forest" or the "Swamp of Marhollow"