  This paper provides a detailed description of the method for the first direct
measurement of all Michel parameters in the $\tau^- \to \mu^-
\bar{\nu}_\mu\nu_\tau$ decay related to the polarization of the daughter muon.
An application of the suggested method in the existing and future experiments
at $e^+e^-$ colliders is considered. We have performed a feasibility study for
the future Super Charm-Tau Factory and Belle II experiments. For the first one,
the sensitivity to the Michel parameters $\xi'$, $\xi''$, $\eta''$,
$\alpha'/A$, and $\beta'/A$ is estimated. For the latter, only one Michel
parameter, $\xi'$, for which the sensitivity is maximum, is considered.
