  1) We introduce random discrete Morse theory as a computational scheme to
measure the complicatedness of a triangulation. The idea is to try to quantify
the frequence of discrete Morse matchings with a certain number of critical
cells. Our measure will depend on the topology of the space, but also on how
nicely the space is triangulated.
  (2) The scheme we propose looks for optimal discrete Morse functions with an
elementary random heuristic. Despite its na\"ivet\'e, this approach turns out
to be very successful even in the case of huge inputs.
  (3) In our view the existing libraries of examples in computational topology
are `too easy' for testing algorithms based on discrete Morse theory. We
propose a new library containing more complicated (and thus more meaningful)
test examples.
