ABSTRACT. Characterizations are given of when the metric projection P_{M} onto a proximinal subspace M has a continuous, pointwise Lipschitz continuous, or Lipschitz continuous selection. Moreover, it is shown that if P_{M} has a continuous selection, then it has one which is also homogeneous and additive modulo M. An analogous result holds if P_{M} has a pointwise Lipschitz or Lipschitz continuous selection provided that M is complemented. If dim M < ∞ and P_{M} is Lipschitz (resp. pointwise Lipschitz) continuous, then P_{M} has a Lipschitz (resp. pointwise Lipschitz) continuous selection. A conjecture of R. Holmes and B. Kripke (Michigan Math. J. 15 (1968), 225–248) is resolved.