Using the theory of nonnegative intersections, duality of the Schubert varieties, and a Pieri-type formula for the varieties of maximal, totally isotropic subspaces, we get an upper bound for the canonical dimension $cd({\rm Spin}_n)$ of the spinor group ${\rm Spin}_n$ . A lower bound is given by the canonical $2$ -dimension $cd_{2}({\rm Spin}_n)$ , computed in [10]. If $n$ or $n+1$ is a power of $2$ , no space is left between these two bounds; therefore, the precise value of $cd({\rm Spin}_n)$ is obtained for such $n$ . We also produce an upper bound for canonical dimension of the semispinor group (giving the precise value of the canonical dimension in the case when the rank of the group is a power of $2$ ) and show that spinor and semispinor groups are the only open cases of the question about canonical dimension of an arbitrary simple split group possessing a unique torsion prime