Omnidirectional Videos (or 360° videos) are widely used in Virtual Reality (VR) to facilitate immersive and interactive viewing experiences. However, the limited spatial resolution in 360° videos does not allow for each degree of view to be represented with adequate pixels, limiting the visual quality offered in the immersive experience. Deep learning Video Super-Resolution (VSR) techniques used for conventional videos could provide a promising software-based solution; however, these techniques do not tackle the distortion present in equirectangular projections of 360° video signals. An additional obstacle is the limited 360° video datasets to study. To address these issues, this paper creates a novel 360° Video Dataset (360VDS) with a study of the extensibility of conventional VSR models to 360° videos. This paper further proposes a novel deep learning model for 360° Video Super-Resolution (360° VSR), called Spherical Signal Super-resolution with a Proportioned Optimisation (S3PO). S3PO adopts recurrent modelling with an attention mechanism, unbound from conventional VSR techniques like alignment. With a purpose-built feature extractor and a novel loss function addressing spherical distortion, S3PO outperforms most state-of-the-art conventional VSR models and 360° specific super-resolution models on 360° video datasets. A step-wise ablation study is presented to understand and demonstrate the impact of the chosen architectural subcomponents, targeted training and optimisation.