  Recently measurements on various spin-1/2 quantum magnets such as
H$_3$LiIr$_2$O$_6$, LiZn$_2$Mo$_3$O$_8$, ZnCu$_3$(OH)$_6$Cl$_2$ and 1T-TaS$_2$
-- all described by magnetic frustration and quenched disorder but with no
other common relation -- nevertheless showed apparently universal scaling
features at low temperature. In particular the heat capacity C[H,T] in
temperature T and magnetic field H exhibits T/H data collapse reminiscent of
scaling near a critical point. Here we propose a theory for this scaling
collapse based on an emergent random-singlet regime extended to include
spin-orbit coupling and antisymmetric Dzyaloshinskii-Moriya (DM) interactions.
We derive the scaling $C[H,T]/T \sim H^{-\gamma} F_q[T/H]$ with $F_q[x] =
x^{q}$ at small $x$, with $q \in$ (0,1,2) an integer exponent whose value
depends on spatial symmetries. The agreement with experiments indicates that a
fraction of spins form random valence bonds and that these are surrounded by a
quantum paramagnetic phase. We also discuss distinct scaling for magnetization
with a $q$-dependent subdominant term enforced by Maxwell's relations.
