  We investigate the role of nuclear spin in elastic scattering of Dark Matter
(DM) neutralinos from nuclei in the framework of the Minimal SUSY standard
model (MSSM). The relative contribution of spin-dependent axial-vector and
spin-independent scalar interactions to the event rate in a DM detector has
been analyzed for various nuclei. Within general assumptions about the nuclear
and nucleon structure we find that for nuclei with atomic weights $A > 50$ the
spin-independent part of the event rate $R_{si}$ is larger than the
spin-dependent one $R_{sd}$ in the domain of the MSSM parameter space allowed
by the known experimental data and where the additional constraint for the
total event rate $R =R_{sd} + R_{si} > 0.01$ is satisfied. The latter reflects
realistic sensitivities of present and near future DM detectors. Therefore we
expect equal chances for discovering the DM event either with spin-zero or with
spin-non-zero isotopes if their atomic weights are $A_{1} \sim A_{2} > 50$.
