  A $k$-tuple $(H_1, \ldots, H_k)$ of core-free subgroups of a finite group $G$
is said to be regular if $G$ has a regular orbit on the Cartesian product
$G/H_1 \times \cdots \times G/H_k$. The regularity number of $G$, denoted
$R(G)$, is the smallest positive integer $k$ with the property that every such
$k$-tuple is regular. In this paper, we develop some general methods for
studying the regularity of subgroup tuples in arbitrary finite groups, and we
determine the precise regularity number of all almost simple groups with an
alternating or sporadic socle. For example, we prove that $R(S_n) = n-1$ and
$R(A_n) = n-2$. We also formulate and investigate natural generalisations of
several well-studied problems on base sizes for finite permutation groups,
including conjectures due to Cameron, Pyber and Vdovin. For instance, we extend
earlier work of Burness, O'Brien and Wilson by proving that $R(G) \leqslant 7$
for every almost simple sporadic group, with equality if and only if $G$ is the
Mathieu group ${\rm M}_{24}$. We also show that every triple of soluble
subgroups in an almost simple sporadic group is regular, which generalises
recent work of Burness on base sizes for transitive actions of sporadic groups
with soluble point stabilisers.
