  We present results from a search for neutrinoless double-$\beta$
($0\nu\beta\beta$) decay using 36.6 g of the isotope $^{150}$Nd with data
corresponding to a live time of 5.25 y recorded with the NEMO-3 detector. We
construct a complete background model for this isotope, including a measurement
of the two-neutrino double-$\beta$ decay half-life of $T^{2\nu}_{1/2}=$[9.34
$\pm$ 0.22 (stat.) $^{+0.62}_{-0.60}$ (syst.)]$\times 10^{18}$ y for the ground
state transition, which represents the most precise result to date for this
isotope. We perform a multivariate analysis to search for \zeronu decays in
order to improve the sensitivity and, in the case of observation, disentangle
the possible underlying decay mechanisms. As no evidence for \zeronu decay is
observed, we derive lower limits on half-lives for several mechanisms involving
physics beyond the Standard Model. The observed lower limit, assuming light
Majorana neutrino exchange mediates the decay, is $T^{0\nu}_{1/2} >$ 2.0
$\times 10^{22}$ y at the 90% C.L., corresponding to an upper limit on the
effective neutrino mass of $\langle m_{\nu} \rangle$ $<$ 1.6 - 5.3 eV..
