We study propagation of the Gabor wave front set for a Schrödinger equation wit ha Hamiltonian that is the Weyl quantization of a quadratic form with nonnegativereal part. We point out that t he singular space associated with the quadratic formplays a crucial role for the understanding of this propagation. We show that the Gaborsingularities of the solution to the equation for positive times are always contained inthe singular space, and that t hey propagate in this set along the flow of the Hamiltonvector field associated with the imaginary part of the quadratic form. As an applicationwe obtain for the heat equation a sufficient condition on the Gabor wave front set of theinitial datum tempered distribution that implies regularization to Schwartz regularityfor positive times.