A compact foliated space with equicontinuous action of the
holonomy pseudogroup can be considered as a topological analogue of a
Riemannian foliation. Molino theory for Riemannian foliations was
generalized for general equicontinuous foliated spaces by Alvarez Lopez
and co-authors. In this talk, we consider a special case when the
foliated space has totally disconnected transversals, and so has the
structure of a lamination. We show that Molino theory extends to this
type of foliated spaces only if the lamination is stable, with the
notion of stability defined in the talk. We introduce algebraic
invariants, which allow us to determine when a lamination is stable, and
give an overview of applications of these invariants in topological and measurable dynamics and in
number theory.