  The importance and signatures of cosmic tau--(anti)neutrinos have been
studied for upward-- and downward--going $\mu^-+\mu^+$ and hadronic shower
event rates relevant for present and future underground water or ice detectors,
utilizing the unique and reliable ultrasmall--$x$ predictions of the dynamical
(radiative) parton model. The upward--going $\mu^- +\mu^+$ event rates
calculated just from cosmic $\nu_{\mu}+\bar{\nu}_{\mu}$ fluxes are sizeably
enhanced by taking into account cosmic $\nu_{\tau}+ \bar{\nu}_{\tau}$ fluxes
and their associated $\tau^- +\tau^+$ fluxes as well. The coupled transport
equations for the upward--going $\stackrel{(-)}{\nu}_{\tau}$ flux traversing
the Earth imply an enhancement of the attenuated and regenerated
$\stackrel{(-)}{\nu}_{\tau}$ flux typically around $10^4-10^5$ GeV with respect
to the initial cosmic flux. This enhancement turns out to be smaller than
obtained so far, in particular for flatter initial cosmic fluxes behaving like
$E_{\nu}^{-1}$. Downward--going $\mu^- +\mu^+$ events and in particular the
background--free and unique hadronic `double bang' and `lollipop' events allow
to test downward--going cosmic $\nu_{\tau} +\bar{\nu}_{\tau}$ fluxes up to
about $10^9$ GeV.
