Quantum-gas microscopes provide direct access to the phases of the Hubbard model, bringing microscopic insight into the complex competition between interactions, SU(2) magnetism, and doping. Alkaline-earth(-like) fermions extend this spin-1/2 paradigm by realizing higher symmetries and giving access to SU($N$) Hubbard models, with rich phase diagrams to be unveiled. Despite its fundamental interest, a microscopic exploration of SU($N$) quantum systems has remained elusive. Here we report the realization of a quantum-gas microscope for fermionic ${}^{87}$Sr. Our imaging scheme, based on cooling and fluorescence on the narrow intercombination line at 689 nm, enables spin-resolved single-atom detection. By implementing a spin-selective optical pumping protocol, we determine the occupation of each of the 10 spin states in a single experimental realization, a crucial capability for probing site-resolved magnetic correlations. We benchmark our method by observing single-particle Larmor precession across the full spin-9/2 ground-state manifold and use numerical simulations of SU($N$) Fermi-Hubbard chains, incorporating the measured detection fidelities, to demonstrate its capability to resolve many-body spin correlations. Our results establish ${}^{87}$Sr quantum-gas microscopy as a powerful approach to study exotic magnetism in the SU($N$) Fermi-Hubbard model, and provide a new detection tool for studies in quantum simulation, computation, and metrology.