Two anomalies at nuclear reactors, one related to the absolute antineutrino flux, one related to the antineutrino spectral shape, have drawn special attention to the field of reactor neutrino physics during the past decade. Numerous experimental efforts have been launched to investigate the reliability of flux models and to explore whether sterile neutrino oscillations are at the base of the experimental findings. This review aims to provide an overview on the status of experimental searches at reactors for sterile neutrino oscillations and measurements of the antineutrino spectral shape in mid-2021. The individual experimental approaches and results are reviewed. Moreover, global and joint oscillation and spectral shape analyses are discussed. Many experiments allow setting of constraints on sterile oscillation parameters, but cannot yet cover the entire relevant parameter space. Others find evidence in favour of certain parameter space regions. In contrast, findings on the spectral shape appear to give an overall consistent picture across experiments and allow narrowing down of contributions of certain isotopes.