  We study regularity properties of frequency measures arising from random
substitutions, which are a generalisation of (deterministic) substitutions
where the substituted image of each letter is chosen independently from a fixed
finite set. In particular, for a natural class of such measures, we derive a
closed-form analytic formula for the $L^q$-spectrum and prove that the
multifractal formalism holds. This provides an interesting new class of
measures satisfying the multifractal formalism. More generally, we establish
results concerning the $L^q$-spectrum of a broad class of frequency measures.
We introduce a new notion called the inflation word $L^q$-spectrum of a random
substitution and show that this coincides with the $L^q$-spectrum of the
corresponding frequency measure for all $q \geq 0$. As an application, we
obtain closed-form formulas under separation conditions and recover known
results for topological and measure theoretic entropy.
