Complex systems are often found to exhibit unexpectedly simple scaling laws that can signal new physical regimes or universal relations between otherwise very different systems. Although this provides a powerful tool for characterising systems close to equilibrium, there are only few known examples where scaling behaviour can be found in dynamical settings. Here we demonstrate power-law scaling in a well-controlled quantum spin system driven out of equilibrium. This enables us to reconstruct the non-equilibrium phase diagram of the system and identify dissipation-dominated, driving-dominated and interaction-dominated regimes. The measured scaling laws show signatures of the underlying ground state phase structure, including paramagnetic behaviour, quantum critical behaviour and a magnetic instability towards states with strong spatial correlations. This opens up a new means to study and classify quantum systems out of equilibrium and extends the domain where scale-invariant behaviour can be found.