Hyperspectral imaging is an important sensing technology with broad applications and impact in areas including environmental science, weather, and geo/space exploration. One important task of hyperspectral image (HSI) processing is the extraction of spectral-spatial features. Leveraging on the recent-developed graph signal processing over multilayer networks (M-GSP), this work proposes several approaches to HSI segmentation based on M-GSP feature extraction. To capture joint spectral-spatial information, we first customize a tensor-based multilayer network (MLN) model for HSI, and define a MLN singular space for feature extraction. We then develop an unsupervised HSI segmentation method by utilizing MLN spectral clustering. Regrouping HSI pixels via MLN-based clustering, we further propose a semi-supervised HSI classification based on multi-resolution fusions of superpixels. Our experimental results demonstrate the strength of M-GSP in HSI processing and spectral-spatial information extraction.