Biswas–Milovic equation with spatio-temporal dispersion serves as a model to study various kinds of imperfections in optical fibres for the solitons propagation. This paper addresses new exact solutions of spatio-temporal Biswas–Milovic equation with Kerr law, dual power law and triple power law nonlinearities. The powerful integration gadget namely, the modified simple equation method is used to construct dark and dark singular solutions along with other traveling wave solutions. The constraint conditions for the perseverance of such solutions fall out naturally.