In this chapter, the nonlinear dynamics and singularity of crossing and product cubic systems with a self-linear and crossing-quadratic product vector field are discussed. The appearing and switching bifurcation dynamics for such product cubic systems are presented through a theorem. The double-inflection saddles, parabola saddles, and third-order parabola saddles exist. The double-inflection saddles are the appearing and switching bifurcations for up- and down-parabola saddles and the appearing bifurcations for matrices of two saddles and two centers. Parabola saddles are the appearing and switching bifurcations of saddle and center. The third-order parabola saddles are (i) the appearing and switching bifurcations of a parabola saddle and a double-inflection saddle, (ii) the appearing bifurcations of three connected parabola saddles, and (iii) the appearing bifurcations of (2×3)-network of simple saddles and centers. The switching bifurcations are obtained through the infinite equilibriums, and they are the inflection source (sink), the parabola-source (sink) infinite equilibriums, and the third-order inflection source (sink) infinite equilibriums.