  We consider low-$x$ behaviour of the spin structure functions $g_1(x)$ and
$h_1(x)$ in the unitarized chiral quark model which combines ideas on the
constituent quark structure of hadrons with a geometrical scattering picture
and unitarity. A nondiffractive singular low-$x$ dependence of $g^p_1(x)$ and
$g_1^n(x)$ indicated by the recent SMC experimental data is described. A
diffractive type smooth behaviour of $h_1(x)$ is predicted at small $x$. The
expectations for the double-spin asymmetries in the low-mass Drell-Yan
production at RHIC in the central region are discussed alongside.
