  In this paper we show that the (un)bounded derived categories$\colon$(i) of
the monomorphism category, (ii) of the morphism category and (iii) of the
double morphism category, admit a periodic infinite ladder of recollements.
These results are based on a characterization that we provide for a recollement
of (compactly generated) triangulated categories to admit a ladder of some
height going either upwards or downwards. Moreover, we introduce and study the
singularity category of the monomorphism category over an Artin algebra
$\Lambda$ and show that there is a periodic infinite ladder that connects this
triangulated category with the standard singularity category of $\Lambda$. We
also provide sufficient conditions for the monomorphism categories of two
algebras to be singularly equivalent. The last aim of this paper is to study
monomorphisms where the domain has finite projective dimension. We show that
the latter category is a Gorenstein subcategory of the monomorphism category
when $\Lambda$ is a Gorenstein Artin algebra. Finally, we consider the category
of coherent functors over the stable category of this Gorenstein subcategory,
and show that it carries a structure of a Gorenstein abelian category.
