Spaceborne synthetic aperture radar (SAR) can provide finely-resolved (meters-scale) images of ocean surface roughness day-or-night in nearly all weather conditions. This makes it a unique asset for many geophysical applications. Initially designed for the measurement of directional ocean wave spectra, Sentinel-1 SAR wave mode (WV) vignettes are small 20 km scenes that have been collected globally since 2014. Recent WV data exploration reveals that many important oceanic and atmospheric phenomena are also well captured, but not yet employed by the scientific community. However, expanding applications of this whole massive dataset beyond ocean waves requires a strategy to automatically identify these geophysical phenomena. In this study, we propose to apply the emerging deep learning approach in ocean SAR scenes classification. The training is performed using a hand-curated dataset that describes ten commonly-occurring atmospheric or oceanic processes. Our model evaluation relies on an independent assessment dataset and shows satisfactory and robust classification results. To further illustrate the model performance, regional patterns of rain and sea ice are qualitatively analyzed and found to be very consistent with independent remote sensing datasets. In addition, these high-resolution WV SAR data can resolve fine, sub-km scale, spatial structure of rain events and sea ice that complement other satellite measurements. Overall, such automated SAR vignettes classification may open paths for broader geophysical application of maritime Sentinel-1 acquisitions.