We investigate laminar Bunsen flames with detailed chemistry and multicomponent transport. The governing equations are discretized by a finite element method on a sequence of adaptively refined, unstructured triangulations. The finite element method is an extension to chemically reacting flows of the streamline diffusion method, including least-squares stabilization of the pressure gradient and the low-Mach continuity equation as well as a shock capturing term designed to control species mass fraction undershoots near flame fronts. Unstructured meshes are adaptively refined based on a posteriori estimates of a user specified functional of the numerical error. These estimates are derived from the dual weighted residual method in the form of element-wise residuals weighted by coefficients depending on the solution of a linearized dual problem that accounts for convective error propagation and multicomponent chemistry couplings. Numerical results are presented to illustrate the efficiency of the proposed methodology and to study the impact of inflow velocity profiles on the structure of several hydrogen-air Bunsen flames