In this paper, we focus on sporadic random-access communications and consider compressed-sensing (CS) techniques to perform the multiuser detection (MUD). Since all the users do not necessarily transmit information, MUD consists in detecting the transmitting users (activity detection) and their corresponding transmitted data (data detection). The main results presented here rely on the exploitation of the user signal alphabet knowledge within the detection step. To this aim, several modifications of the group orthogonal matching pursuit (GOMP) algorithm were proposed, differing in the way the modulation alphabet knowledge is considered within the detection stage. These modifications can be extended to any greedy-based CS-MUD. To overcome the error floor occurring at high SNR with a higher number of active users, we then propose an iterative minimization-based MUD algorithm that alternates between activity and data detection. Compared to the existing GOMP-based CS-MUD, the proposed modified GOMP algorithms exhibit a significant gain with almost the same complexity. The iterative minimization-based MUD algorithm has a higher complexity but outperforms all the others without any observed error-floor.