We consider doubly selective multiuser/multiple-input-multiple-output (MIMO) channel estimation and data detection using superimposed training. The time- and frequency-selective fading channel is assumed to be well described by a discrete prolate spheroidal basis expansion model (DPS-BEM) using Slepian sequences as basis functions. A user-specific periodic (nonrandom) training sequence is arithmetically added (superimposed) at low power to each user's information sequence at the transmitter before modulation and transmission. A two-step approach is adopted, where, in the first step, we estimate the channel using only the first-order statistics of the observations. In this step, however, the unknown information sequence acts as interference, resulting in a poor signal-to-noise ratio (SNR). We then iteratively reduce the interference in the second step by employing an iterative channel-estimation and data-detection approach, where, by utilizing the detected symbols from the previous iteration, we sequentially improve the multiuser/MIMO channel estimation and symbol detection. Simulation examples demonstrate that, without incurring any transmission data rate loss, the proposed approach is superior to the conventional time-multiplexed (TM) training for uncoordinated users, where the multiuser interference in channel estimation cannot be eliminated and is competitive with the TM training for coordinated users, where the TM training design allows for multiuser-interference-free channel estimation.