Research about singularities has been driving scientific advancements across mathematics and physics. Comprehending and harnessing the novel properties of singularities in photonics can facilitate the development of integrated micro-nano devices in diverse platforms. Herein, we provide a comprehensive overview of photonic singularities emerging in structured light fields and metamaterial structures. We classify them into several representative types: real-space singularities, momentum-space singularities, and parameter-space singularities, with discussions of their intriguing topological and dynamical properties. Moreover, we report on the latest applications of photonic singularities in broad areas, ranging from light routing, lasing, sensing, and optical manipulation to imaging and display. This review connects the singularity phenomena in different photonic systems, bridging the abstract concepts with emerging practical applications. It underscores the significance of photonic singularities in both fundamental science and various on-chip applications.