Free convection occuring during the drying of plane layers of polymer solutions may be due to various mechanisms, based on buoyancy or Marangoni effect, of thermal or solutal origin. This theoretical and numerical work provides all the required tools to analyze thoroughly the problem. In this transient flow, different methods (frozen time, non-normal, nonlinear methods) are proposed to predict critical times for convection onset and threshold values for convection. Nonlinear and non-normal methods give similar results, within the uncertainty inherent to any transient problem. It is shown that, when linear stability analysis indicates the presence of several instability mechanisms, it is necessary to invoke nonlinear arguments to establish the leading mechanism. The proposed methodology is then applied to experimental results from the literature for two polymer solutions (Polyisobutylene/toluene and Polystyrene/toluene).