Disordered potentials fundamentally affect transport and coherence in quantum systems, giving rise to a Bose-glass phase in interacting bosonic systems—an insulating yet compressible phase lacking long-range coherence. Directly measuring a reduced coherence length of the Bose glass has been a outstanding challenge. We address this by employing Talbot interferometry combined with single-atom-resolved detection in a quantum-gas microscope. Using ultracold bosonic atoms in a two-dimensional lattice with site-resolved, reproducible disorder, we identify the Bose-glass phase through in situ density distributions and particle-number fluctuations, quantified via the Edwards-Anderson parameter, and through the visibility of interference patterns after time of flight. By driving the system across the Bose-glass phase, we further observe signatures of nonergodic dynamics. Our studies provide a starting point to further explore disordered systems in and out of equilibrium, and are relevant for understanding the dynamics and stability of disordered and glasslike quantum states in solid-state systems.