  We report the result for a search for the leptonic decay of $B^+ \to \mu^+ \,
\nu_{\mu}$ using the full Belle data set of 711 fb${}^{-1}$ of integrated
luminosity at the $\Upsilon(4S)$ resonance. In the Standard Model leptonic
$B$-meson decays are helicity and CKM suppressed. To maximize sensitivity an
inclusive tagging approach is used to reconstruct the second $B$ meson produced
in the collision. The directional information from this second $B$ meson is
used to boost the observed $\mu$ into the signal $B$ meson rest-frame, in which
the $\mu$ has a monochromatic momentum spectrum. Though its momentum is smeared
by the experimental resolution, this technique improves the analysis
sensitivity considerably. Analyzing the $\mu$ momentum spectrum in this frame
we find $\mathcal{B}(B^+ \to \mu^+ \, \nu_\mu) = \left( 5.3 \pm 2.0 \pm 0.9
\right) \times 10^{-7}$ with a one-sided significance of 2.8 standard
deviations over the background-only hypothesis. This translates to a
frequentist upper limit of $\mathcal{B}(B^+ \to \mu^+ \, \nu_{\mu}) < 8.6
\times 10^{-7}$ at 90% CL. The experimental spectrum is then used to search for
a massive sterile neutrino, $B^+ \to \mu^+ \, N$, but no evidence is observed
for a sterile neutrino with a mass in a range of 0 - 1.5 GeV. The determined
$B^+ \to \mu^+ \, \nu_{\mu}$ branching fraction limit is further used to
constrain the mass and coupling space of the type II and type III
two-Higgs-doublet models.
