Viscoelastic flows are characterised by fast spatial and temporal variations in the solution featuring thin stress boundary near walls and stress concentrations in the vicinity of geometrical singularities. Resolving these fast variations of the fields in space and time is important for two reasons: (i) they affect the quantity of interest of the computation (e.g. drag force); and (ii) they are commonly believed to be associated with the numerical breakdown of the computation. Traditional discretisation methods such as finite differences or low-order finite elements require a large number of degrees of freedom to resolve these variations. Spectral methods enable this issue to be resolved by defining spatial expansions that are able to represent such variations with a smaller number of degrees of freedom. However, such methods are limited in terms of geometric flexibility. Recently, the spectral/hp element method (Karniadakis and Sherwin, 2005) has been developed in order to guarantee both spectral convergence, and geometric flexibility by allowing the use of quadrilateral and triangular elements. Our work is the first attempt to apply this method to viscoelastic free surface flows in arbitrary complex geometries. The conservation equations are solved in combination with the Oldroyd-B or Giesekus constitutive equation using the DEVSS-G/DG formulation. The combination of this formulation with a spectral element method is novel. A continuous approximation is employed for the velocity and discontinuous approximations for pressure, velocity gradient and polymeric stress. The conservation equations are discretised using the Galerkin method and the constitutive equation using a discontinuous Galerkin method to increase the stability of the approximation. The viscoelastic free surface is traced using an arbitrary Lagrangian Eulerian method. The performance of our scheme is demonstrated on the time-dependent Poiseuille flow in a channel, the flow around a cylinder and the die-swell problem.