  The harmonic measure $\nu$ on the boundary of the group $Sol$ associated to a
discrete random walk of law $\mu$ was described by Kaimanovich. We investigate
when it is absolutely continuous or singular with respect to Lebesgue measure.
By ratio entropy over speed, we show that any countable non-abelian subgroup
admits a finite first moment non-degenerate $\mu$ with singular harmonic
measure $\nu$. On the other hand, we prove that some random walks with finitely
supported step distribution admit a regular harmonic measure. Finally, we
construct some exceptional random walks with arbitrarily small speed but
singular harmonic measures. The two later results are obtained by comparison
with Bernoulli convolutions, using results of Erd\"os and Solomyak.
