Synthetic quantum systems with interacting constituents play an important role in quantum information processing and in elucidating fundamental phenomena in many-body physics. Following impressive advances in cooling and trapping techniques, ultracold molecular gases have emerged as a promising quantum platform with several advantageous properties. These include a large set of internal states for encoding quantum information, long coherence times and long-range interactions. The latter are expected to allow the exploration of intriguing phases of correlated matter, such as topological superfluids, quantum spin liquids and fractional Chern insulators. Probing correlations in these phases is crucial to understand their microscopic properties, necessitating the development of new experimental techniques. I will present our work on measuring real and momentum-space correlations in molecular gases with single-particle resolution. In particular, I will focus on measurements of non-equilibrium correlations in various quantum spin models, such as spin-exchange or anisotropic Heisenberg models. These experiments push the frontier of probing and controlling interacting systems of ultracold molecules, with prospects for characterizing entangled states useful for quantum computation and metrology.