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We consider social resource allocations that deliver an array of scarce supports to a diverse population. Such allocations pervade social service delivery, such as provision of homeless services, assignment of refugees to cities, among others. At issue is whether allocations are fair across sociodemographic groups and intersectional identities. Our paper shows that necessary trade-offs exist for fairness in the context of scarcity; many reasonable definitions of equitable outcomes cannot hold simultaneously except under stringent conditions. For example, defining fairness in terms of improvement over a baseline inherently conflicts with defining fairness in terms of loss compared with the best possible outcome. Moreover, we demonstrate that the fairness trade-offs stem from heterogeneity across groups in intervention responses. Administrative records on homeless service delivery offer a real-world example. Building on prior work, we measure utilities for each household as the probability of reentry into homeless services if given three homeless services. Heterogeneity in utility distributions (conditional on received services) for several sociodemographic groups (e.g. single women with children versus without children) generates divergence across fairness metrics. We argue that such heterogeneity, and thus, fairness trade-offs pervade many social policy contexts.