A pseudo-solid domain mapping technique coupled with the DEVSS finite element formulation is applied to study the effects of viscoelasticity on free surface flows. Two distinct flow types are analyzed: the flow induced by a long air bubble steadily displacing a polymeric liquid confined by two parallel plates, i.e. Hele-Shaw flow, and the slot coating of viscoelastic fluids in the low metering rate limit. The Oldroyd-B, FENE-CR, and FENE-P constitutive equations are used to model the viscoelastic fluid. Our study reveals the formation of an elastic boundary layer in the capillary-transition region near the bubble front at moderate Weissenberg numbers while the stress field in the parallel flow region remains largely unaffected by the dynamics of the free surface. Our calculations show that the increase in the hydrodynamic coating thickness due to viscoelasticity often reported in planar injection experiments [Physica A 220 (1995) 60; J. Non-Newtonian Fluid Mech. 71 (1997) 73] is associated with the onset of these elastic boundary layers and is strongly dependent on the physical properties of the coating fluid.