Cognitive Radio and the emerging Non-Orthogonal Multiple Access (NOMA) techniques hold the potential to fulfill the increasing demands of radio spectrum by 5G-and-B5G wireless systems. It is a usual practice in underlay cognitive radio networks to take assistance from intermediate relays to reach the remote destination. Putting together, NOMA based Cognitive Relay Networks (NCRNs) have recently gained tremendous research attention to improve the spectrum utilization efficiency. This article studies the application of best relay selection (BRS) in downlink scenario of NCRNs in which a base station (BS) being unable to communicate directly with the far user U 2 takes assistance from the near user U 1 and from the best Decode-and-Forward (DF) relay selected from the potential NCRN operating in underlay environment. Three BRS schemes are proposed selecting a relay which (i) maximizes signal-to-noise ratio (SNR) on relay-U 2 link, (ii) minimizes interference on relay-primary user (PU) link, (iii) maximizes the quotient of relay-U 2 link's SNR and relay-PU link's interference power. Scheme 3 considers the existence of U 1 − U 2 link as well whereas the other two schemes do not. To characterize the performance of the proposed schemes assuming Rayleigh channel model, closed-form expressions of probability distribution function (PDF) of received SNR at U 2 , average number of reliable relays, outage probability and bit error rate are obtained. For insight analysis, analytical results are validated through simulations which reveal that relay selection in NCRNs is different from non-cognitive networks and is feasible in low-to-medium SNR regions. The signal from near user further improved the outage performance.