  We use a strong-disorder renormalization group (SDRG) method and ground-state
quantum Monte Carlo (QMC) simulations to study S=1/2 spin chains with random
couplings, calculating disorder-averaged spin and dimer correlations. The QMC
simulations demonstrate logarithmic corrections to the power-law decaying
correlations obtained with the SDRG scheme. The same asymptotic forms apply
both for systems with standard Heisenberg exchange and for certain multi-spin
couplings leading to spontaneous dimerization in the clean system. We show that
the logarithmic corrections arise in the valence-bond (singlet pair) basis from
a contribution that can not be generated by the SDRG scheme. In the model with
multi-spin couplings, where the clean system dimerizes spontaneously, random
singlets form between spinons localized at domain walls in the presence of
disorder. This amorphous valence-bond solid is asymptotically a random-singlet
state and only differs from the random-exchange Heisenberg chain in its
short-distance properties.
