We address uplink user admission control problems in multiuser wireless systems with multiple single-antenna users and a multi-antenna receiver. Our aim is to select the maximum number of users that can simultaneously satisfy the given quality of service (QoS) and power constraints from a large number of candidates in the system. Both sum and per-user transmit power constraints are considered. By utilizing uplink-downlink duality, we formulate the uplink user admission problems into sparsity-maximization problems, which are NP-hard. Inspired by compressive sensing techniques, we propose novel methods to tackle them by first using convex relaxation and then applying fast iterative methods on the convex-relaxed solutions. Simulations show that the proposed algorithms achieve excellent performance. The numbers of admitted users are close to the optimum ones obtained by exhaustive search when the QoS requirement is medium or low. Compared to a greedy algorithm, the proposed methods can admit up to 24% more users in the simulations.