The accuracy and quality of approximation of solutions to viscoelastic flows in complex geometries depends strongly on using numerically stable methods that are computationally costs effective. In practical applications of viscoelastic flow modeling, stress and pressure singularities arise, resulting in layers that are difficult to resolve. We present Finite Element based a posteriori error estimates for viscoelastic flows governed by differential constitutive laws. These estimates are constructed based on a general nonlinear residual-based framework for a posteriori error estimation. We examine two main discretizations: one with continuous stress approximation and another with a discontinuousstress approximation. Using these estimators as error indicators and the DEVSS formulation of the govening equations we then perform adaptive computation of viscoelastic flows We examine the channel flow with an obstacle using a pseudo time stepping technique (θ-method).