  Sextic double solids, double covers of $\mathbb P^3$ branched along a sextic
surface, are the lowest degree Gorenstein Fano 3-folds, hence are expected to
behave very rigidly in terms of birational geometry. Smooth sextic double
solids, and those which are $\mathbb Q$-factorial with ordinary double points,
are known to be birationally rigid. In this article, we study sextic double
solids with an isolated compound $A_n$ singularity. We prove a sharp bound $n
\leq 8$, describe models for each $n$ explicitly and prove that sextic double
solids with $n > 3$ are birationally non-rigid.
