In many experimental measurements, corrupted data and outliers can significantly distort the coherent structures identified through traditional modal analysis techniques. This distortion becomes particularly pronounced at higher frequencies, where the corresponding modes are more susceptible to contamination from measurement noise and uncertainties. To address these limitations, we introduce a novel approach, robust spectral proper orthogonal decomposition (robust SPOD), which incorporates the robust principal component analysis method by Scherl et al. (2020) into the SPOD framework by Towne et al. (2018). In this work, we assess robust SPOD effectiveness through applications to two distinct fluid dynamics problems: a numerically simulated turbulent subsonic jet flow field and experimental data of the flow within an open cavity from Zhang et al. (2019). When applied to turbulent jet data artificially corrupted by salt and pepper and Gaussian noise, the robust SPOD produces more converged and physically interpretable modes than the standard SPOD method. Furthermore, we illustrate how robust SPOD can be employed as a powerful tool for data denoising, relying on signal reconstruction from denoised modes. The analysis of the open cavity flow with the robust SPOD yields smoother spatial distributions of modes, particularly at high frequencies and for higher-order modes when compared to the conventional SPOD approach.