This preprint is not a finished product. It is presently intended to gather community feedback. We investigate the problem of recovering jointly $r$-rank and $s$-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that $m \asymp r s \ln(en/s)$ measurements make the recovery possible in theory, meaning via a nonpractical algorithm. For arbitrary measurements, we also show that practical recovery could be achieved when $m \asymp r s \ln(en/s)$ via an iterative-hard-thresholding algorithm provided one could answer positively a question about head projections for the jointly low-rank and bisparse structure. Some related questions are partially answered in passing. For the rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements.