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We investigate the role of nuclear spin in elastic scattering of Dark Matter(DM) neutralinos from nuclei in the framework of the Minimal SUSY standardmodel (MSSM). The relative contribution of spin-dependent axial-vector andspin-independent scalar interactions to the event rate in a DM detector hasbeen analyzed for various nuclei. Within general assumptions about the nuclearand nucleon structure we find that for nuclei with atomic weights $A > 50$ thespin-independent part of the event rate $R_{si}$ is larger than thespin-dependent one $R_{sd}$ in the domain of the MSSM parameter space allowedby the known experimental data and where the additional constraint for thetotal event rate $R =R_{sd} + R_{si} > 0.01$ is satisfied. The latter reflectsrealistic sensitivities of present and near future DM detectors. Therefore weexpect equal chances for discovering the DM event either with spin-zero or withspin-non-zero isotopes if their atomic weights are $A_{1} \sim A_{2} > 50$.Comment: 14 pages, 5 .ps-figure, Preprint JINR E2-93-44

Direct detection of supersymmetric dark matter and the role of the target nucleus spin