Let L be any finite simple lattice of at least three elements, whose co-atoms intersect to 0. One principal result of the paper is that L is not dual isomorphic to the lattice of subvarieties of any locally finite variety. A second principal result is that these statements are equivalent: (i) L is isomorphic to the congruence lattice of a finite algebra with one basic operation; (ii) L is isomorphic either to the subspace lattice of a finite vector space, or for some permutation σ of a finite domain, to the lattice of equivalence relations invariant under σ.