The critical conditions leading to fracture in elongation and different types of flow instabilities were examined in uniaxial elongation and in a capillary rheometer equipped with dies having different entry profiles. Either ductile or brittle fracture may be observed, ductile being related to necking of material. The critical stress approach was used to predict fracture in elongation. All linear polymers studied in this work exhibited ductile fracture in uniaxial elongation, but the transition to brittle fracture is discussed in relation to existing experiments with other materials. In a ductile fracture regime, critical stress and work both increase with an increasing rate of deformation, whereas in a brittle regime the critical values remain constant. The converging flow studies indicated that two types of flow instability that have been previously related to each other, namely, pressure oscillations and volume distortions, are of different origins. The critical flow rate for pressure oscillations is independent of entry profile, and the origin for this type of instability lies along the wall of the capillary. On the other hand, the critical flow rate for volume distortions increased with a decreasing entry angle, indicating that volume distortions are not a consequence of pressure oscillation, nor are their origin at the capillary wall. Numerical simulations were used to determine the stress profiles within the flow, and it was shown that the onset of volume distortions is directly related to the magnitude of elongational stress and work, and may therefore be considered to be caused by fracture in elongation. In dies with 90° entry profile, volume distortions were observed simultaneously with pressure oscillations, making it difficult to distinguish between the two phenomena.