The number of active users, their timing offsets, and their (possibly dispersive) channels with the access point are performance-critical parameters for wireless code division multiple access (CDMA). Estimating them as accurately as possible using as short as possible training sequences can markedly improve error performance as well as the capacity of CDMA systems. The fresh look advocated here permeates benefits from recent advances in variable selection and compressive sampling approaches to multiuser communications by casting estimation of these parameters as a sparse linear regression problem. Novel estimators are developed by exploiting two forms of sparsity present: the first emerging from user (in)activity, and the second from the uncertainty on user delays and channel taps. Simulated tests demonstrate a large gain in performance when sparsity-aware estimators of CDMA parameters are compared to sparsity-agnostic standard least-squares-based alternatives.