Effective cloud and cloud shadow detection is a critical prerequisite for accurate retrieval of concentrations of atmospheric methane (CH4) or other trace gases in hyperspectral remote sensing. This challenge is especially pertinent for MethaneSAT, a satellite mission launched in March 2024, to fill a significant data gap in terms of resolution, precision and swath between coarse-resolution global mappers and fine-scale point-source imagers of methane, and for its airborne companion mission, MethaneAIR. MethaneSAT delivers hyperspectral data at an intermediate spatial resolution (approx. 100 x 400, m), whereas MethaneAIR provides even finer resolution (approx. 25 m), enabling the development of highly detailed maps of concentrations that enable quantification of both the sources and rates of emissions. In this study, we use machine learning methods to address the cloud and cloud shadow detection problem for sensors with these high spatial resolutions. Cloud and cloud shadows in remote sensing data need to be effectively screened out as they bias methane retrievals in remote sensing imagery and impact the quantification of emissions. We deploy and evaluate conventional techniques-including Iterative Logistic Regression (ILR) and Multilayer Perceptron (MLP)-with advanced deep learning architectures, namely U-Net and a Spectral Channel Attention Network (SCAN) method. Our results show that conventional methods struggle with spatial coherence and boundary definition, affecting the detection of clouds and cloud shadows. Deep learning models substantially improve detection quality: U-Net performs best in preserving spatial structure, while SCAN excels at capturing fine boundary details... Our data and code is publicly available at: https://doi.org/10.7910/DVN/IKLZOJ