We present results from a search for neutrinoless double-$beta$ ($0 ubetabeta$) 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 $T^{2 u}_{1/2}=$[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 $T^{0 u}_{1/2} >$ 2.0 $times 10^{22}$ y at the 90% C.L., corresponding to an upper limit on the effective neutrino mass of $langle m_{ u} rangle$ $<$ 1.6 - 5.3 eV..
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