  We describe a procedure, called regularisation, that allows us to study
geometric structures on Lie algebroids via foliated geometric structures on a
manifold of higher dimension. This procedure applies to various classes of Lie
algebroids; namely, those whose singularities are of b^k, complex-log, or
elliptic type, possibly with self-crossings.
  One of our main applications is a proof of the Weinstein conjecture for
overtwisted b^k-contact structures. This was proven by Miranda-Oms using a
certain technical hypothesis. Our approach avoids this assumption by reducing
the proof to the foliated setting. As a by-product, we also prove the Weinstein
conjecture for other Lie algebroids.
  Along the way we also introduce tangent distributions, i.e. subbundles of Lie
algebroids, as interesting objects of study and present a number of
constructions for them.
