Are there regions of space from which no light can escape (trapping horizons)?: almost certainly. Supermassive black holes can have large horizons, with surprisingly small space-time curvature, and we understand gravity and GR well enough that we can be reasonably sure that such horizons exist. Are there singularities inside these horizons? - almost certainly not. Physicists dislike singularities, which is one reason why they search for a quantum theory of gravity. So the question of what lies inside a black hole can only be answered when someone comes up with a consistent quantum theory of gravity.
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