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. A number of relays (i.e., $K$ relays) are uniformly distributed within the disc. A pair of RS schemes are considered insightfully: 1) Single-stage RS (SRS) scheme; and 2) Two-stage RS (TRS) scheme. In order to characterize the performance of these two RS schemes, new closed-form expressions for both exact and asymptotic outage probabilities are derived. Based on analytical results, the diversity orders achieved by the pair of RS schemes for FD/HD cooperative NOMA are obtained. Our analytical results reveal that: i) The FD-based RS schemes obtain a zero diversity order, which is due to the influence of loop interference (LI) at the relay; and ii) The HD-based RS schemes are capable of achieving a diversity order of $K$, which is equal to the number of relays. Finally, simulation results demonstrate that: 1) The FD-based RS schemes have better outage performance than HD-based RS schemes in the low signal-to-noise radio (SNR) region; 2) As the number of relays increases, the pair of RS schemes considered are capable of achieving the lower outage probability; and 3) The outage behaviors of FD/HD-based NOMA SRS/TRS schemes are superior to that of random RS (RRS) and orthogonal multiple access (OMA) based RS schemes.