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Network Utility Maximization (NUM) is a mathematical framework that has endowed researchers with powerful methods for designing and analyzing classical communication protocols. NUM has also enabled the development of distributed algorithms for solving the resource allocation problem, while at the same time providing certain guarantees, e.g., that of fair treatment, to the users of a network. We extend here the notion of NUM to quantum networks, and propose three quantum utility functions -- each incorporating a different entanglement measure. We aim both to gain an understanding of some of the ways in which quantum users may perceive utility, as well as to explore structured and theoretically-motivated methods of simultaneously servicing multiple users in distributed quantum systems. Using our quantum NUM constructions, we develop an optimization framework for networks that use the single-photon scheme for entanglement generation, which enables us to solve the resource allocation problem while exploring rate-fidelity tradeoffs within the network topologies that we consider. We learn that two of our utility functions, which are based on distillable entanglement and secret key fraction, are in close agreement with each other and produce similar solutions to the optimization problems we study. Our third utility, based on entanglement negativity, has more favorable mathematical properties, and tends to place a higher value on the rate at which users receive entangled resources, compared to the two previous utilities, which put a higher emphasis on end-to-end fidelity. These contrasting behaviors thus provide ideas regarding the suitability of quantum network utility definitions to different quantum applications.