Time-resolved x-ray microscopy is used in a low-alpha synchrotron operation mode to image spin dynamics at an unprecedented combination of temporal and spatial resolution. Thereby, nanoscale spin waves with wavelengths down to 70 nm and frequencies up to 30 GHz are directly observed in ferromagnetic thin film microelements with spin vortex ground states. In an antiparallel ferromagnetic bilayer system, we detect the propagation of both optic and acoustic modes, the latter exhibiting even a strong non-reciprocity. In single-layer systems, quasi-uniform spin waves are observed together with modes of higher order (up to the 4th order), bearing precessional nodes over the thickness of the film. Furthermore, the effects of magnetic material properties, film thickness, and magnetic fields on the spin-wave spectrum are determined experimentally. Our experimental results are consistent with numerical calculations from a micromagnetic theory even on these so-far unexplored time- and length scales.