Some Loci of Rational Cubic Fourfolds

26 pages, with an Appendix by Giovanni Staglian\`o. Comments welcomeIn this paper we investigate the geography of some codimension two loci inside the moduli space of smooth cubic hypersurfaces in $\mathbb{P}^5$ and the rationality of their elements. In particular, we study the loci of cubics which contain a plane and another surface, whose ideal is generated by quadratic equations. This is the case of cubic and quartic scrolls and Veronese surfaces - and some of their degenerations. Using the fact that all degenerations of quartic scrolls in $\mathbb{P}^5$ contained in a smooth cubic hypersurface are surfaces with one apparent double point, we also show that every cubic hypersurface belonging to the divisor $\mathcal{C}_{14}$ in the moduli space is rational