Abstract The morphological development of wrinkles along the surface of a Bunsen flame in a weakly-turbulent flow is investigated, with turbulence added solely to disturb the flame front. The resulting flame-flow interactions are examined using a hybrid Navier–Stokes/front-tracking methodology within the context of the hydrodynamic theory. Topological markers based on the skewness of curvature are introduced to distinguish between sub- and super-critical conditions, or the absence/presence of the Darrieus–Landau instability, respectively. We show that for sub-critical conditions disturbances created along the flame surface are dampened when convected downstream along the flame front, and the flame surface is only weakly perturbed. For super-critical conditions, on the other hand, disturbances of the flame front are amplified when advected downstream leading to a highly corrugated surface and a flame brush of increasing thickness. A measure of these dramatic changes is included in the mean local stretch rate which, when properly modulated by a Markstein length, directly affects the turbulent flame speed.