  We prove a transverse diameter theorem in the context of Lorentzian
foliations, which can be interpreted as a Hawking--Penrose-type singularity
theorem for timelike geodesics transverse to the foliation. In order to develop
the necessary machinery we introduce and study a novel causality structure on
the leaf space via the transverse Lorentzian geometry on the foliated manifold.
We describe the initial rungs of a transverse causal ladder and relate them to
their standard counterparts on an underlying foliated spacetime. We show how
these results can be interpreted as doing Lorentzian (and more generally
semi-Riemannian) geometry on low-regularity spaces that can be realized as leaf
spaces of foliations. Accordingly, we discuss how all of these concepts and
results apply to Lorentzian orbifolds, insofar as these can be seen as leaf
spaces of a specific class of Lorentzian foliations. In particular, we derive
an associated Lorentzian timelike diameter theorem on orbifolds.
