Dual-function radar communications (DFRC) systems have recently been proposed to enable the coexistence of radar and wireless communications, which in turn alleviates the increased spectrum congestion crisis. In this paper, we consider the problem of sparse transmit array design for DFRC systems by antenna selection where same or different antennas are assigned to different functions. We consider three different types of DFRC systems which implement different simultaneous beamformers associated with single and different sparse arrays with shared aperture. We utilize the array configuration as an additional spatial degree of freedom (DoF) to suppress the cross-interference and facilitate the cohabitation of the two system functions. It is shown that the use of sparse arrays adds to improved angular resolution with well-controlled sidelobes on DFRC system paradigm. The utilization of sparse arrays in DFRC systems is validated using simulation examples.