We use methods of singularity theory to classify the local geometry of the discontinuity set, together with associated local dynamics, for a discrete dynamical system that represents a natural class of oscillator with one degree of freedom impacting against a fixed obstacle. We also include descriptions of the generic transitions that occur in the discontinuity set as the position of the obstacle is smoothly varied. The results can be applied to any choice of restitution law at impact. The analysis provides a general setting for the study of local and global dynamics of discontinuous systems of this type, for example giving a geometric basis for the possible construction of Markov partitions in certain cases.