  Kodaira fibrations are surfaces of general type with a non-isotrivial
fibration, which are differentiable fibre bundles. They are known to have
positive signature divisible by $4$. Examples are known only with signature 16
and more. We review approaches to construct examples of low signature which
admit two independent fibrations. Special attention is paid to ramified covers
of product of curves which we analyse by studying the monodromy action for
bundles of punctured curves.
  As a by-product we obtain a classification of all fix-point-free
automorphisms on curves of genus at most $9$.
