A broad class of soil fungi form the annular patterns known as 'fairy rings' and provide one of the only means to observe spatio-temporal dynamics of otherwise cryptic fungal growth processes in natural environments. We present observations of novel spiral and rotor patterns produced by fairy ring fungi and explain these behaviors mathematically by first showing that a well known model of fairy ring fungal growth and the Gray-Scott reaction-diffusion model are mathematically equivalent. We then use bifurcation analysis and numerical simulations to identify the conditions under which spiral waves and rotors can arise. We demonstrate that the region of dimensionless parameter space supporting these more complex dynamics is adjacent to that which produces the more familiar fairy rings, and identify experimental manipulations to test the transitions between these spatial modes. These same manipulations could also feasibly induce fungal colonies to transition from rotor/spiral formation to a set of richer, as yet unobserved, spatial patterns.