We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. This equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These solutions are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces H^s(T^2) (0 < s < 1). More precisely, we construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the H^s topology and whose H^s norm can grow by any given factor. In my talk, I will also say some words about the ongoing work concerning Sobolev instability of more general 2D quasi-periodic solutions. The subject of this talk is partly motivated by the problem of infinite energy cascade for 2D NLS, and it is a joint work with M. Guardia, Z. Hani, A. Maspero and M. Procesi.