Any Riemannian manifold M admits at each point a neighbourhood over which exist polar-like coordinates, namely normal coordinates. Assuming given a subset X of M which is not submanifold, we can nevertheless equip its smooth part with the restriction of the ambient Riemannian structure and try to understand the behaviour of geodesics nearby any non smooth point. The most expected occurrence of such situation is when M is an affine or projective space (real or complex) and X is an affine or projective variety with non-empty singular locus. The standard strategy is to use a parameterization of X (resolution of singularities) in such a way that the source space is a manifold with boundary (mapped surjectively onto the singular locus) and pull back the Riemannian structure onto this manifold, using this parameterization, and work nearby the boundary with a degenerate tensor along the boundary. In a joint work with D. Grieser (Univ. Oldenburg, Germany) we discuss the problem of an exponential-like map at the singular point of a class of isolated surface singularities of an Euclidean space, called cuspidal surface, which are explicit in some sense. I will state the trichotomy of this class of surface regarding the existence and the injectivity of an exponential-like mapping at the singular point of this class of surface... and explain a bit if times allows.