  We use numerical techniques to study dynamical properties at finite
temperature ($T$) of the Heisenberg spin chain with random exchange couplings,
which realizes the random singlet (RS) fixed point in the low-energy limit.
Specifically, we study the dynamic spin structure factor $S(q,\omega)$, which
can be probed directly by inelastic neutron scattering experiments and, in the
limit of small $\omega$, in nuclear magnetic resonance (NMR) experiments
through the spin-lattice relaxation rate $1/T_1$. Our work combines three
complementary methods: exact diagonalization, matrix-product-state algorithms,
and stochastic analytic continuation of quantum Monte Carlo results in
imaginary time. Unlike the uniform system, whose low-energy excitations at low
$T$ are restricted to $q$ close to $0$ and $\pi$, our study reveals a
continuous narrow band of low-energy excitations in $S(q,\omega)$, extending
throughout the Brillouin zone. Close to $q=\pi$, the scaling properties of
these excitations are well captured by the RS theory, but we also see
disagreements with some aspects of the predicted $q$-dependence further away
from $q=\pi$. Furthermore we find spin diffusion effects close to $q=0$ that
are not contained within the RS theory but give non-negligible contributions to
the mean $1/T_1$. To compare with NMR experiments, we consider the distribution
of the local $1/T_1$ values, which is broad, approximately described by a
stretched exponential. The mean value first decreases with $T$, but starts to
increase and diverge below a crossover temperature. Although a similar
divergent behavior has been found for the static uniform susceptibility, this
divergent behavior of $1/T_1$ has never been seen in experiments. Our results
show that the divergence of the mean $1/T_1$ is due to rare events in the
disordered chains and is concealed in experiments, where the typical $1/T_1$
value is accessed.
