This paper investigates the multiuser diversity introduced by opportunistic user selection in cognitive radio (CR) networks, where multiple cognitive users request to access the spectral resources of the licensed (primary) user. We investigate a simple cognitive user selection strategy aiming at maximizing the received signal-to-interference-plus-noise ratio (SINR) for a given power budget, under interference constraints to the primary. We study the statistics of the SINR at the cognitive receiver, and derive exact analytical expressions of its probability density function (PDF). We then analytically calculate the diversity gains introduced in the system due to the selection of one cognitive user amongst multiple candidates compared to the case when only one cognitive user exists and no selection occurs. Furthermore, we utilize the PDF of the SINR to predict the bit error rate (BER) of the selected cognitive user. Finally, the asymptotic behavior of the diversity gains for the low transmit power region of the primary and cognitive links, and as the number of candidate links becomes large is also investigated. All three multiaccess scenarios are investigated, namely multiple access channel (MAC), broadcast channel (BC) and parallel access channel (PAC), and the results show that the analytically derived expressions closely match simulated performance.