  We study the low-energy physics of a broad class of time-reversal invariant
and SU(2)-symmetric one-dimensional spin-S systems in the presence of quenched
disorder via a strong-disorder renormalization-group technique. We show that,
in general, there is an antiferromagnetic phase with an emergent SU(2S+1)
symmetry. The ground state of this phase is a random singlet state in which the
singlets are formed by pairs of spins. For integer spins, there is an
additional antiferromagnetic phase which does not exhibit any emergent symmetry
(except for S=1). The corresponding ground state is a random singlet one but
the singlets are formed mostly by trios of spins. In each case the
corresponding low-energy dynamics is activated, i.e., with a formally infinite
dynamical exponent, and related to distinct infinite-randomness fixed points.
The phase diagram has two other phases with ferromagnetic tendencies: a
disordered ferromagnetic phase and a large spin phase in which the effective
disorder is asymptotically finite. In the latter case, the dynamical scaling is
governed by a conventional power law with a finite dynamical exponent.
