Flame dynamics involve many effects (hydrodynamics, acoustics, thermal expansion, reaction and more), and they are known to display unstable behaviour in a large variety of configurations. These instabilities can be underpinned by rich interactions, for instance between shear mechanisms, flame front dynamics and thermo-acoustic radiation. I will visit three very different scenarios where instabilities manifest themselves, and examine these with linear analysis:

1. a laminar slot flame under external excitation,

2. a V-shaped flame issuing from an annular nozzle,

3. a conical Bunsen flame of premixed hydrogen.

For the slot flame example, I will critically discuss the concept of flame transfer functions, and compare it against resolvent analysis. The V-flame displays a subcritical bifurcation due to intrinsic thermo-acoustic instability. In the hydrogen flame, linear eigenmode analysis predicts that a sub-unity Lewis number leads to the onset of « polyhedral » flame shapes.