Bounds are derived for the acoustic losses such that a thermoacoustic system with a given flame can be guaranteed to be stable. The analysis is based on the flame’s acoustic input-to-output properties represented by its scattering matrix. The developed analytical and numerical techniques allow estimating the maximum reflection coefficients (equivalently – acoustic losses) which are sufficient to ensure stable operation of a given burner. It is shown that the calculated numerical upper-bound is less conservative than the analytical one. The frequency dependence of the required acoustic losses provides (i) a thermo-acoustic signature of the flame and (ii) guidelines for the proper design of the up- and downstream acoustics from the flame. The method is illustrated on two burners/flames of premixed multiple Bunsen type. The frequency dependence of the upper bounds allows to identify those frequency ranges where the flame is more likely to cause instability of the complete system.