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 [1]. 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.