DBSCAN is an algorithm that performs clustering in the presence of noise. In this paper, we provide two constructions that allow DBSCAN to be implemented neuromorphically, using spiking neural networks. The first construction is termed "flat," resulting in large spiking neural networks that compute the algorithm quickly, in five timesteps. Moreover, the networks allow pipelining, so that a new DBSCAN calculation may be performed every timestep. The second construction is termed "systolic", and generates much smaller networks, but requires the inputs to be spiked in over several timesteps, column by column. We provide precise specifications of the constructions and analyze them in practical neuromorphic computing settings. We also provide an open-source implementation.