Quasicrystals are aperiodically ordered structures with unconventional rotational symmetry. Their peculiar features have been explored in photonics to engineer bandgaps for light waves. Magnons (spin waves) are collective spin excitations in magnetically ordered materials enabling non–charge-based information transmission in nanoscale devices. Here, we report on a two-dimensional magnonic quasicrystal formed by aperiodically arranged nanotroughs in ferrimagnetic yttrium iron garnet. By phase-resolved spin wave imaging at gigahertz frequencies, multidirectional emission from a microwave antenna is evidenced, allowing for a quasicontinuous radial magnon distribution, not observed in reference measurements on a periodic magnonic crystal. We observe partial forbidden gaps, which are consistent with analytical calculations and indicate band formation as well as a modified magnon density of states due to backfolding at pseudo-Brillouin zone boundaries. The findings promise as-desired filters and magnonic waveguides reaching out in a multitude of directions of the aperiodic lattice.