  Let $G$ be an almost simple sporadic group and let $H$ be a soluble subgroup
of $G$. In this paper we prove that there exists $x,y \in G$ such that $H \cap
H^x \cap H^y=1$, which is equivalent to the bound $b(G,H) \leqslant 3$ with
respect to the base size of $G$ on the set of cosets of $H$. This bound is best
possible. In this setting, our main result establishes a strong form of a more
general conjecture of Vdovin on the intersection of conjugate soluble subgroups
of finite groups. The proof uses a combination of computational and
probabilistic methods.
