The plastics industry today sees huge wastage through product defects caused by unstable flows during the manufacturing process. In addition, many production lines are throughput-limited by a flow speed threshold above which the process becomes unstable. Therefore, it is critically important to understand the mechanisms behind these instabilities. In order to investigate the flow of a molten plastic, the first step is a model of the liquid itself, a relation between its current stress and its flow history called a constitutive relation. These are derived in many ways and tested on several benchmark flows, but rarely is the stability of the model used as a criterion for selection. The relationship between the constitutive model and the stability properties of even simple flows is not yet well understood. We show that in one case a small change to the model, which does not affect the steady flow behaviour, entirely removes a known instability. In another, a change that makes a qualitative difference to the steady flow makes only tiny changes to the stability. The long-term vision of this research is to exactly quantify what are the important properties of a constitutive relation as far as stability is concerned. If we could understand that, not only could very simple stability experiments be used to choose the best constitutive models for a particular material, but our ability to predict and avoid wasteful industrial instabilities would also be vastly improved.