  The paper is concerned with random $S$-adic systems arising from an i.i.d.
sequence of unimodular substitutions. Using equidistribution results of Benoist
and Quint, we show in Theorem 3.3 that, under some natural assumptions, if the
Lyapunov exponent of the spectral cocycle is strictly less that 1/2 of the
Lyapunov exponent of the random walk on $SL(2,\mathbb{R})$ driven by the
sequence of substitution matrices, then almost surely the spectrum of the
$S$-adic $\mathbb{Z}$-action is singular with respect to any (fixed in advance)
continuous measure. Finally, the appendix discusses the weak-mixing property
for random $S$-adic systems associated to the family of substitutions
introduced in Example 4.2.
