  We re-examine the lower bound on the mass of the Higgs boson, $M_H$, from
Standard Model vacuum stability including next-to-leading-log radiative
corrections. This amounts to work with the full one-loop effective potential,
$V(\phi)$, improved by two-loop RGE, and allows to keep control of the scale
invariance of $V$ in a wide range of the $\phi$-field. Our results show that
the bound is ${\cal O}\ (10\ GeV)$ less stringent than in previous estimates.
In addition we perform a detailed comparison between the SM lower bounds on
$M_H$ and the supersymmetric upper bounds on it. It turns out that depending on
the actual value of the top mass, $M_t$, the eventually measured Higgs mass can
discard the pure SM, the Minimal Supersymmetric Standard Model or both.
