  A colouring of a graph G is called distinguishing if its stabiliser in Aut G
is trivial. It has been conjectured that, if every automorphism of a locally
finite graph moves infinitely many vertices, then there is a distinguishing
2-colouring. We study properties of random 2-colourings of locally finite
graphs and show that the stabiliser of such a colouring is almost surely
nowhere dense in Aut G and a null set with respect to the Haar measure on the
automorphism group. We also investigate random 2-colourings in several classes
of locally finite graphs where the existence of a distinguishing 2-colouring
has already been established. It turns out that in all of these cases a random
2-colouring is almost surely distinguishing.
