Problems with efficient solution except for a small fraction of inputs The halting problem for Turing machines is perhaps the canonical undecidable
  set. Nevertheless, we prove that there is an algorithm deciding almost
  all instances of it. The halting problem is therefore among the growing collection
  of those exhibiting the “black hole” phenomenon of complexity theory,
  by which the difficulty of an unfeasible or undecidable problem is confined
  to a very small region, a black hole, outside of which the problem is easy.  
  
  [Joel David Hamkins and Alexei Miasnikov, "The halting problem is decidable on a set of asymptotic probability one", 2005]


Can anyone provide references to other “black holes” in complexity theory, or another place where this or related concepts are discussed?
I'm not sure whether this is what you're looking, but the phase transition in random SAT is an example. Let $\rho$ be the ratio of number of clauses to number of variables. Then a random SAT instance with parameter $\rho$ is very likely to be satisfiable if $\rho$ is less than a fixed constant (near 4.2) and is very likely to be unsatisfiable if $\rho$ is a little bit more than this constant. The "black hole" is the phase transition.