  In this work we study the Lepton Flavour Violating semileptonic $\tau$
decays: 1) $\tau \to \mu PP$ with $PP= \pi^+\pi^-, \pi^0\pi^0, K^+K^-, K^0
{\bar K}^0$; 2) $\tau \to \mu P$ with $P=\pi^0, \eta, \eta'$ and 3) $\tau \to
\mu V$ with $V = \rho^0, \phi$. We work within the context of two constrained
MSSM scenarios: the CMSSM and NUHM. A full one-(SUSY)loop computation is
presented and the importance of the various contributions, the $\gamma$-, $Z$-,
and Higgs bosons mediated ones are analysed. The hadronization of quark
bilinears is performed within the chiral framework. Some discrepancies in the
predicted rates for BR($\tau \to \mu \eta$), BR($\tau \to \mu \eta'$) and
BR($\tau \to \mu K^+K^-$) are found with respect to previous estimates, which
will be commented here. These three channels will be shown to be the most
competitive ones to test simultaneously the Lepton Flavour Violation and the
Higgs sector. We further present here a set of approximate formulas for all the
semileptonic channels which we believe can be useful for further comparison
with present and future data.
