We introduce non-Hermitian plasmonic waveguide-cavity systems with topological edge states (TESs) at singular points. The compound unit cells of the structures consist of metal-dielectric-metal (MDM) stub resonators side-coupled to an MDM waveguide. We show that we can realize both a TES and an exceptional point at the same frequency when a proper amount of loss is introduced into a finite three-unit-cell structure. We also show that the finite structure can exhibit both a TES and a spectral singularity when a proper amount of gain is introduced into the structure. In addition, we show that we can simultaneously realize a unidirectional spectral singularity and a TES when proper amounts of loss and gain are introduced into the structure. We finally show that this singularity leads to extremely high sensitivity of the reflected light intensity to variations of the refractive index of the active materials in the structure. TESs at singular points could potentially contribute to the development of singularity-based plasmonic devices with enhanced performance.