The Tutte polynomial of a graph, also known as the partition function of the q -state Potts model, is a 2-variable polynomial graph invariant of considerable importance in both combinatorics and statistical physics. It contains several other polynomial invariants, such as the chromatic polynomial and flow polynomial as partial evaluations, and various numerical invariants, such as the number of spanning trees, as complete evaluations. We have developed the most efficient algorithm to-date for computing the Tutte polynomial of a graph. An important component of the algorithm affecting efficiency is the choice of edge to work on at each stage in the computation. In this paper, we present and discuss two edge-selection heuristics which (respectively) give good performance on sparse and dense graphs. We also present experimental data comparing these heuristics against a range of others to demonstrate their effectiveness.