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This article analyses the transient viscoelastic response of strain-hardening fluids in filament stretching flows. We utilise an Arbitrary Lagrangian/Eulerian temporal approach (ALE), coupled with a particle-tracking procedure for free-surface movement and a hybrid finite volume/element method upon the domain. We are able to contrast findings between Oldroyd, Giesekus and linear Phan-Thien/Tanner models, and distinguish between single and multi-mode implementations. In this manner, we identify the impact that greater severe strainhardening has in this transient flow context. Contributions from shear-thinning rheology may be gathered in particular by comparing single-mode solution response between a shearthinning Giesekus and a constant shear viscosity Oldroyd-B model. A parameter study on inertial and surface tension effects has been undertaken, where we isolate the occurrence of asymmetries in the flow under certain conditions, leading to the onset and formulation of bead-like structures. This elucidates the specific localised influence that surface tension and gravitational forces have upon some stretching filament flows.