We study the stable rationality problem for quadric and cubic surface bundles over surfaces from the point of view of the specialization method for the Chow group of 0-cycles. Our main result is that a very general hypersurface X of bidegree (2, 3) in is not stably rational. Via projections onto the two factors, is a cubic surface bundle and is a conic bundle, and we analyze the stable rationality problem from both these points of view. Also, we introduce, for any $$n\geqslant 4$$ n⩾4 , new quadric surface bundle fourfolds with discriminant curve of degree 2n, such that $$X_n$$ Xn has nontrivial unramified Brauer group and admits a universally $$\mathrm {CH}_0$$ CH0 -trivial resolution.