This study is concerned with the investigation of the associated properties of subcell discretisations for viscoelastic flows, where aspects of compatibility of solution function spaces are paramount. We introduce one new scheme, through a subcell finite element approximation fe(sc), and compare and contrast this against two precursor schemes - one with finite element discretisation in common, but at the parent element level quad-fe; the other, at the subcell level appealing to hybrid finite element/finite volume discretisation fe/fv(se). To conduct our comparative study, we consider Oldroyd modelling and two classical steady benchmark flow problems to assess issues of numerical accuracy and stability-cavity flow and contraction flow. We are able to point to specific advantages of the finite element subcell discretisation and appreciate the characteristic properties of each discretisation, by analysing stress and flow field structure up to critical states of Weissenberg number. Findings reveal that the subcell linear approximation for stress within the constitutive equation (either fe or fv) yields a more stable scheme, than that for its quadratic counterpart (quad-fe), whilst still maintaining second-third order accuracy. The more compatible form of stress interpolation within the momentum equation is found to be via the subcell elements under fe(sc); yet, this makes no difference under fe/fv(sc). Furthermore, improvements in solution representation are gathered through enhanced upwinding forms, which may be coupled to stability gains with strain-rate stabilisation. (C) 2007 Elsevier B.V. All rights reserved.