The application of optimisation approaches to the spatial design of urban water systems can result in unequal distributions of infrastructure services among different communities unless spatial equity objectives are explicitly considered. Differences between urban communities can be exacerbated if the co-benefits of water infrastructure are not properly distributed over the urban catchment. This is especially important when designing sustainable urban drainage systems, which carry several co-benefits in addition to the attenuation of stormwater runoff, such as the improvement of urban landscape and mental health of the residents. In this work, we propose a new multi-objective optimisation framework that takes into account both cost-effectiveness and socio-economic aspects of urban drainage infrastructure design. The proposed framework considers the minimisation of capital costs, average flood damages, and total suspended solids as design objectives, as well as the minimisation of inequalities in the spatial distribution of flood damages and green infrastructure benefits quantified via appropriate equity metrics. A population-based multi-objective optimisation method is linked to a hydrologic-hydraulic model to determine a set of Pareto-optimal design portfolios associated with different trade-offs between traditional drainage design objectives and social goals. The proposed multi-criteria design approach was applied to a synthetic case study where sustainable urban drainage systems are used to expand the drainage capacity of an existing pipe network. The results demonstrate the applicability of the proposed methodology to the design of urban drainage systems and provide insights into the trade-offs between overall cost-effectiveness and social equity in urban infrastructure design decisions.