  The muon transverse polarization in the $K^+ \leftarrow \pi^0 \mu^+ \nu_\mu$
decay will be measured at the $10^{-4}$ level in forthcoming experiments. We
compare the phenomenological perspectives with the theoretical predictions in
supersymmetric extensions of the standard model. In the minimal extension,
CP-violating phases lead to a non-zero transverse polarization, that however is
too small to account for a positive experimental signal. The problems that one
encounters when departing from minimal assumptions are discussed. An observable
effect is possible if the hypothesis of R-parity conservation is relaxed, but
only at the price of assuming a very special pattern for the R-parity breaking
couplings.
