We study here some theoretical model problems, with the goal of obtaining a better understanding of viscoelastic free surface flows and their unique flow instabilities. The first analysis examines the stability of inward radial flow of an Oldroyd-B fluid in a washer-shaped domain, showing that the azimuthal compression in this flow leads to a free surface instability -- a crinkling or buckling in the azimuthal direction. This instability is suppressed by surface tension at large wavenumbers, and growth rates are strongly attenuated by solvent viscosity. A second analysis shows how thin stress boundary layers can develop in free surface flows, and the final analysis takes the stress localization idea literally, using as a model of a stress boundary layer a thin elastic membrane. Under elongation at constant enclosed volume, initially axisymmetric membranes with fixed circular ends, e.g. truncated cones, become unstable with respect to nonaxisymmetric disturbances, again due to azimuthal compressive stresses. The resulting configurations appear similar to those observed in filament stretching experiments.