  It is rather well-known that spacetime singularities are not covariant under
field redefinitions. A manifestly covariant approach to singularities in
classical gravity was proposed in arXiv:2008.09387. In this paper, we start to
extend this analysis to the quantum realm. We identify two types of covariant
singularities in field space corresponding to geodesic incompleteness and
ill-defined path integrals (hereby dubbed functional singularities). We argue
that the former might not be harmful after all, whilst the latter makes all
observables undefined. We show that the path-integral measure is regular in any
four-dimensional theory of gravity without matter or in any theory in which
gravity is either absent or treated semi-classically. This might suggest the
absence of functional singularities in these cases, however it can only be
confirmed with a thorough analysis, case by case, of the path integral. We
provide a topological and model-independent classification of functional
singularities using homotopy groups and we discuss examples of theories with
and without such singularities.
