Dimers of metal nanospheres are well-known for their characteristic anisotropic optical response. Here, we demonstrate in single-particle SERS experiments that individual gold trimers and 3D superstructures exhibit a polarization-independent SERS response. This optical behavior of single particle clusters provides constant SERS signals, independent of the mutual orientation of the incident laser polarization and the plasmonic nanostructure, which is desired or even required in many SERS applications.