Optimization with stochastic dominance constraints has recently received an increasing amount of attention in the quantitative risk management literature. Instead of requiring that the probabilistic description of the uncertain parameters be exactly known, this paper presents the first comprehensive study of a data-driven formulation of the distributionally robust second-order stochastic dominance constrained problem (DRSSDCP) that hinges on using a type-1 Wasserstein ambiguity set. This formulation allows us to identify solutions with finite sample guarantees and solutions that are asymptotically consistent when observations are independent and identically distributed. It is furthermore for the first time shown to be axiomatically motivated in an environment with distribution ambiguity. Leveraging recent results in the field of robust optimization, we further formulate the DRSSDCP as a multistage robust optimization problem, and further propose a tractable conservative approximation that exploits finite adaptability and a scenario-based lower bounding problem, both of which can reduce to linear programs under mild conditions. We then propose the first exact optimization algorithm for this DRSSDCP, which efficiency is confirmed by our numerical results. Finally, we illustrate how the data-driven DRSSDCP can be applied in practice on resource allocation problems with both synthetic and real data. Our empirical results show that with a proper adjustment of the size of the Wasserstein ball, DRSSDCP can reach acceptable out-of-sample feasibility while generating strictly better performance than what is achieved by the reference strategy.