We prove for a large class of compact metric spaces, including those manifolds of dimension at least two, Hilbert cube manifolds, and homogeneous Menger manifolds, that ‘most’ self-homeomorphisms (in the sense of residual set of homeomorphisms) have certain properties. Specifically, if F: X → X is one of these homeomorphisms, then F admits
• a dense, open wandering set;
• a nowhere dense chain recurrent set;
• an infinite collection of attractors (and repellers), each of which has nonempty interior and cannot be reduced to a ‘smallest’ attractor (or ‘largest’ repeller); and an uncountable collection of pairwise disjoint quasi-attractors.
We also discuss the topology of the boundaries of attractors.