The origin of deterministic macroscopic properties often lies in microscopic stochastic motion. Magnetic fluctuations that manifest as domain avalanches and chaotic magnetization jumps exemplify such stochastic motion and have been studied in great detail. Here we report Fourier space studies of avalanches in a system exhibiting competing magnetic stripe and skyrmion phase using a soft X-ray speckle metrology technique. We demonstrate the existence of phase boundaries and underlying critical points in the stripe and skyrmion phases. We found that distinct scaling and universality classes are associated with these domain topologies. The magnitude and frequency of abrupt magnetic domain jumps observed in the stripe phase are dramatically reduced in the skyrmion phase. Our results provide an incisive way to probe and understand phase stability in systems exhibiting complex spin topologies.