Maintaining and modulating mechanical anisotropy is essential for biological processes. However, how this is achieved at the microscopic scale in living soft matter is not always clear. Although Brillouin light scattering (BLS) spectroscopy can probe the mechanical properties of materials, spatiotemporal mapping of mechanical anisotropies in living matter with BLS microscopy has been complicated by the need for sequential measurements with tilted excitation and detection angles. Here we introduce Brillouin light scattering anisotropy microscopy (BLAM) for mapping high-frequency viscoelastic anisotropy inside living cells. BLAM employs a radial virtually imaged phased array that enables the collection of angle-resolved dispersion in a single shot, thus enabling us to probe phonon modes in living matter along different directions simultaneously. We demonstrate a precision of 10 MHz in the determination of the Brillouin frequency shift, at a spatial resolution of 2 mu m. Following proof-of-principle experiments on muscle myofibres, we apply BLAM to the study of two fundamental biological processes. In plant cell walls, we observe a switch from anisotropic to isotropic wall properties that may lead to asymmetric growth. In mammalian cell nuclei, we uncover a spatiotemporally oscillating elastic anisotropy correlated to chromatin condensation. Our results highlight the role that high-frequency mechanics can play in the regulation of diverse fundamental processes in biological systems. We expect BLAM to find diverse applications in biomedical imaging and material characterization.