What is the morphology of a black hole?

Are black holes thought to be spherical, flat, or something else, something perhaps more interesting? What physical space would the largest supermassive black hole fill? How far is the black hole itself from its event horizon?

This question continues the Black Hole Series. Previous entries included:
1  —  “Can a black hole overeat?”
2  —  “How big is your average black hole?”
3  —  “What happens when a black hole evaporates?”
4  —  “Would an ordinary black hole attract antimatter?”
5  —  “Could you kill an ordinary black hole by feeding it antimatter?”

Also, @mattbrowne asked What exactly happens when a (hypothetical) antimatter black hole merges with a ‘normal’ black hole? shortly before I arrived here. That excellent question deserves to be part of the series. Feel free to add any other notable black hole questions from whatever date.