This paper examines how surface deformations affect the stability of a slowly evaporating solvent-polymer mixture. The destabilizing effect of surface-tension variations arising from evaporation-induced concentration gradients and the counteracting influence of mean gravity and surface tension are incorporated into the mathematical model. A linear stability analysis that takes advantage of the separation between the characteristic time scales of the slowly evolving base state and the perturbations is carried out in combination with numerical solutions of the linearized system. It is shown that the onset of instability can occur for Marangoni numbers that are much lower than the critical value for a non-deformable surface. Moreover, two types of Marangoni instabilities appear in the system: one is associated with the traditional stationary instability, and the other is an oscillatory instability that is not present for a non-deformable liquid surface. A region of the parameter space where the oscillatory instability dominates is identified and used to formulate appropriate conditions for future experiments. © 2014 The authors 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.