  The resummation of $O(\alpha_s^{l+1} \ln^{2l} x)$ terms in the evolution
equation of the singlet part of $g_1(x,Q^2)$ is carried out. The corresponding
singlet evolution kernels are calculated explicitely. The leading small-$x$
contribution to the three-loop splitting function matrix is determined in the
$\overline{\rm MS}$ scheme. Relations are derived for the case of ${\cal N} =
1$ supersymmetric Yang--Mills field theory. Numerical results are presented for
the polarized singlet and gluon densities, and the structure functions $g_1^{\,
p}(x,Q^2)$ and $g_1^{\, n}(x,Q^2)$. They are compared for different assumptions
on the non--perturbative input distributions, and the stability of the results
against presently unknown subleading contributions is investigated.
