  Measurement of the transverse muon polarization $P^{\bot}_{\mu}$ in the $K^+
\rightarrow \mu^+\nu\gamma$ decay will be attempted for the first time at the
ongoing KEK E246 experiment and also at a proposed BNL experiment. We provide a
general analysis of how $P^{\bot}_{\mu}$ is sensitive to the physical
$CP$-violating phases in new physics induced four-Fermi interactions, and then
we calculate the dominant contributions to $P^{\bot}_{\mu}$ from squark family
mixings in generic supersymmetric models. Estimates of the upper bounds on
$P^{\bot}_{\mu}$ are also given. It is found that a supersymmetry-induced
right-handed quark current from $W$ boson exchange gives an upper limit on
$P^{\bot}_{\mu}$ as large as a few per cent, whereas with
charged-Higgs-exchange induced pseudoscalar interaction, $P^{\bot}_{\mu}$ is no
larger than a few tenths of a per cent. Possible correlations between the muon
polarization measurements in $K^+ \rightarrow \mu^+\nu\gamma$ and $K^+
\rightarrow \pi^0\mu^+\nu$ decays are discussed, and distinctive patterns of
this correlation from squark family-mixings and from the three-Higgs-doublet
model are noted.
