  We prove that any ergodic measure-preserving action of an irreducible lattice
in a semisimple group, with finite center and each simple factor having rank at
least two, either has finite orbits or has finite stabilizers. The same
dichotomy holds for many commensurators of such lattices.
  The above are derived from more general results on groups with the Howe-Moore
property and property $(T)$. We prove similar results for commensurators in
such groups and for irreducible lattices (and commensurators) in products of at
least two such groups, at least one of which is totally disconnected.
