Real-space singularities govern a broad spectrum of wave phenomena, yet they remain largely unexplored in elastic wave systems. Here, we report hybrid topological singularities that emerge on the surfaces of finite-sized solids due to the full vectorial character of elastic waves. These textures fuse spin-field singularities with displacement-field singularities and exhibit unique non-pairwise topological charge dynamics. Moreover, a subset of these singularities imprint dislocation lines onto the radiated acoustic field, generating robust acoustic vortices in free space from an otherwise achiral source and structure. Our results establish elastic waves as a powerful platform for engineering real-space singularities and open avenues for singular phononics and the exploration of rich topological defects in elastic media.