Is the random attractor a singleton?

In analogy with deterministic theory, one initially expects that random attractors may be complex objects, but in a number of examples where the corresponding deterministic dynamics has a non-trivial attractor it has been proved that the random attractor is indeed a random singleton. This phenomenon is related to stochastic bifurcation and its different interpretations. One-point random attractors seem to be more common than expected; but there are also examples where it is not one point. We shall review a few old and new facts about this topic, and related issues as the properties of the two-point motion.