  Induced by quenched disorder, random-singlet states occur in a variety of
low-dimensional spin-1/2 antiferromagnets, some of them candidates for quantum
spin liquids. Here we report measurements of the specific heat, magnetization,
and magnetic susceptibility of nanocrystalline Sr21Bi8Cu2(CO3)2O41, a
quasi-one-dimensional spin-1/2 antiferromagnet with alternating bonds. The
results uncover the predominant presence of random-singlet spin pairs in this
material, with a logarithmic probability distribution, P(J), of the
renormalized, emergent exchange interaction, J, in zero magnetic field and P(J)
proportional to 1/J in magnetic fields. We postulate that these unexpected J
dependences, in contrast to the usual P(J) \propto 1/$J^\gamma$ with 0 <
$\gamma$ < 1, and possibly also the dichotomy, arise from the finite size of
the nanocrystals. Scaling functions for the specific heat and magnetization
reproduce our magnetocaloric-effect data, with no adjustable parameters.
