Recently Yu. Bilu, P. Habegger and L. K\"uhne proved that no singular modulus
can be a unit in the ring of algebraic integers. In this paper we study for
which sets S of prime numbers there is no singular modulus that is an S-units.
Here we prove that when the set S contains only primes congruent to 1 modulo 3
then no singular modulus can be an S-unit. We then give some remarks on the
general case and we study the norm factorizations of a special family of
singular moduli.Comment: Version changed according to the referee's comments. The final
version appears in Manuscripta Mathematica,
https://doi.org/10.1007/s00229-020-01230-
