This paper investigates the impact of relay selection (RS) on the performance of cooperative non-orthogonal multiple access (NOMA), where relays are capable of working in either full-duplex (FD) or half-duplex (HD) mode. The locations of relays considered in the networks are modeled using stochastic geometry. The single-stage relay selection (SRS) scheme is proposed. In order to characterize the performance of SRS scheme, new closed-form expressions for both exact and asymptotic outage probabilities are derived. Based on the analytical results, the diversity orders achieved by the SRS scheme for FD/HD cooperative NOMA are obtained. It is confirmed that the FD-based SRS schemes obtain a zero diversity order, while the HD-based RS is capable of achieving a diversity order of K. Simulation results show that the outage performance of FD-based RS scheme outperforms HD-based RS scheme in the low signal-to-noise radio (SNR) region rather than in the high SNR region.