  The wavevector and temperature dependent static spin susceptibility,
\chi_s(Q,T), of clean interacting Fermi systems is considered in dimensions
1\leq d \leq 3. We show that at zero temperature \chi_s is a nonanalytic
function of |Q|, with the leading nonanalyticity being |Q|^{d-1} for 1<d<3, and
Q^2\ln|Q| for d=3. For the homogeneous spin susceptibility we find a
nonanalytic temperature dependence T^{d-1} for 1<d<3. We give qualitative
mode-mode coupling arguments to that effect, and corroborate these arguments by
a perturbative calculation to second order in the electron-electron interaction
amplitude. The implications of this, in particular for itinerant
ferromagnetism, are discussed. We also point out the relation between our
findings and established perturbative results for 1-d systems, as well as for
the temperature dependence of \chi_s(Q=0) in d=3.
