This paper reports work on a nonpremixed half-dump combustor, in which methane is injected at the backward-facing step, and mixes and burns with the air flowing past the step in the unsteady recirculation zone. The flow and geometric parameters are widely varied, to gradually change from conditions of low-amplitude noise to excitation of high-amplitude discrete tones. The purpose of the work is to focus on the transition from the former condition to the latter, and to mark the onset of instability. Dimensionless groups such as the Helmholtz and Strouhal numbers are formed based on the observed dominant frequencies, whose variation with the air flow Reynolds number is used to identify the oscillations as those due to the natural acoustic modes or the vortex shedding process. High-speed chemiluminescence imaging reveals shedding of vortical structures in the flame zone. With variation in the conditions, flow-acoustic lock-on and transition from one vortex shedding mode to another is marked by nonlinearity in the corresponding amplitude variations. Such conditions are identified as the onset of instability in terms of the ratio of the flow time scale to the acoustic time scale and mapped against the operating fuel-air equivalence ratio of the combustor.