  Inspired by the recent theoretical suggestion that the random-bond $S=1/2$
antiferromagnetic Heisenberg model on the triangular and the kagome lattices
might exhibit a randomness-induced quantum spin liquid (QSL) behavior when the
strength of the randomness exceeds a critical value, and that this
"random-singlet state" might be relevant to the QSL behaviors experimentally
observed in triangular organic salts $\kappa {\rm -(ET)_2 Cu_2 (CN)_3}$and
${\rm EtMe_3 Sb[Pd(dmit)_2]_2}$ and in kagome herbertsmithite ${\rm
CuZn_3(OH)_6Cl_2}$, we further investigate the nature of the static and the
dynamical spin correlations of these models. We compute the static and the
dynamical spin structure factors, $S({\bf q})$ and $S({\bf q},\omega)$, by
means of an exact diagonalization method. In both triangular and kagome models,
the computed $S({\bf q},\omega)$ in the random-singlet state depends on the
wavevector ${\bf q}$ only weakly, robustly exhibiting gapless behaviors
accompnied by the broad distribution extending to higher energy $\omega$.
Especially in the strongly random kagome model, $S({\bf q},\omega)$ hardly
depends on ${\bf q}$, and exhibits an almost flat distribution for a wide range
of $\omega$, together with a $\omega=0$ peak. These features agree
semi-quantitatively with the recent neutron-scattering data on a single-crystal
herbertsmithite, suggesting that the QSL state observed in herbersmithite might
indeed be the randomness-induced QSL state, {\it i.e.\/}, the random-singlet
state.
