We report on theoretical and experimental investigations on spontaneous and stimulated Brillouin scattering during operation of a high-power single-frequency polarization-maintaining ytterbium doped fiber amplifier. For different amplifier configurations with co- and counter-propagating seed and pump radiation the evolution of Brillouin scattering spectra was investigated with a heterodyne detection scheme. Spontaneous Brillouin gain spectra at low powers were additionally investigated using a pump-probe technique. The data obtained from these experiments have been compared with a theoretical model based on coupled intensity equations. A Brillouin scattering suppression has been investigated theoretically and experimentally with externally applied temperature gradients along the fiber resulting in up to 3.5 dB suppression and 115 W of amplifier output power.