This article investigates the impacts of relay selection schemes on cooperative underlay cognitive radio non-orthogonal multiple access networks, where the partial relay selection scheme, the max–min relay selection scheme and the two-stage relay selection scheme are applied in the network. Moreover, decode-and-forward protocol is used at the transmission relays. What’s more, in order to show the effect of the schemes on the considered network, the closed-form expressions and asymptotic expressions for the outage probability of the system are derived. Furthermore, the outage performance under the effect of perfect and imperfect successive interference cancellation is analysed. Numerical results are given to illustrate the impacts of the relay selection schemes, the number of relays, the residual interference factor and the power allocation factor on the outage performance. Finally, Monte Carlo simulations are presented to validate the accuracy of the numerical results.