Sub-GeV dark matter (DM) particles produced via thermal freeze-out evade many of the strong constraints on heavier DM candidates but at the same time face a multitude of new constraints from laboratory experiments, astrophysical observations and cosmological data. In this work we combine all of these constraints in order to perform frequentist and Bayesian global analyses of fermionic and scalar sub-GeV DM coupled to a dark photon with kinetic mixing. For fermionic DM, we find viable parameter regions close to the dark photon resonance, which expand significantly when including a particle-antiparticle asymmetry. For scalar DM, the velocity-dependent annihilation cross section evades the strongest constraints even in the symmetric case. Using Bayesian model comparison, we show that both asymmetric fermionic DM and symmetric scalar DM are preferred over symmetric fermionic DM due to the reduced fine-tuning penalty. Finally, we explore the discovery prospects of near-future experiments both in the full parameter space and for specific benchmark points. We find that the most commonly used benchmark scenarios are already in tension with existing constraints and propose a new benchmark point that can be targeted with future searches.