Manin's conjecture predicts the quantitative behaviour of rational points on algebraic varieties. For a primitive positive definite quadratic form Q with integer coefficients, the equation x^3 = Q(y)z represents a class of singular cubic hypersurfaces. In this paper, we mainly introduce the distribution of rational points on these hypersurfaces, and describe the ideas, methods, and some results. This is a joint work with Jie Wu and Jianya Liu.