We use direct numerical simulation (DNS) of the Navier–Stokes equations in the low-Mach-number limit to investigate the hydrodynamic instability of a lifted jet diffusion flame. We obtain steady solutions for flames using a finite rate reaction chemistry, and perform a linear global stability analysis around these steady flames. We calculate the direct and adjoint global modes and use these to identify the regions of the flow that are responsible for causing oscillations in lifted jet diffusion flames, and to identify how passive control strategies might be used to control these oscillations. We also apply a local stability analysis to identify the instability mechanisms that are active. We find that two axisymmetric modes are responsible for the oscillations. The first is a high-frequency mode with wavemaker in the jet shear layer in the premixing zone. The second is a low-frequency mode with wavemaker in the outer part of the shear layer in the flame. We find that both of these modes are most sensitive to feedback involving perturbations to the density and axial momentum. Using the local stability analysis, we find that the high-frequency mode is caused by a resonant mode in the premixing region, and that the low-frequency mode is caused by a region of local absolute instability in the flame, not by the interaction between resonant modes, as proposed in Nichols et al. (Phys. Fluids, vol. 21, 2009, article 015110). Our linear analysis shows that passive control of the low-frequency mode may be feasible because regions up to three diameters away from the fuel jet are moderately sensitive to steady control forces.