Recently, singular spectrum analysis (SSA) and two-dimensional SSA (2D-SSA) have been utilized to successfully extract features from hyperspectral images (HSIs) in the spectral and spatial domains, respectively. However, the limitations of both algorithms are that they cannot extract joint spectral-spatial features, and the data they reconstruct can result in over-smoothness, leading to inaccurate classification. To address these issues, this study proposes a novel local and global three-dimensional SSA (LG-3D-SSA) algorithm that spatially partition the hyperspectral cube into subcubes of the same size. Unlike the previous two versions, 3D development of SSA (3D-SSA) employs an embedding cubic window and is applied to each spatial partition separately. This approach allows for extracting spectral-spatial features using local information from each partition. Furthermore, a post-processing step is proposed to utilize global information between partitions with majority class samples. Experimental data on three publicly-available datasets indicate that the proposed method outperforms other state-of-the-art techniques in terms of classification accuracy.