Summary: The problem of a flame stabilized by a line source of fuel of strength \(2\pi\) \(\kappa\) is solved numerically. As \(\kappa\) is varied, first a bifurcation from an axisymmetric solution to a stationary cellular solution is found. Increasing \(\kappa\) further, a sequence of transitions to stationary cellular solutions of increasing mode number is found. Each transition is accompanied by a region of bistability where two stable stationary cellular solutions coexist, each with its own domain of attraction. Evidence is presented to show that the modal transitions occur via subcritical bifurcations.