The structure of the concentration field of a decaying substance produced by chemical sources and advected by a smooth incompressible two-dimensional flow is investigated. We focus our attention on the nonuniformities of the Hölder exponent of the resulting filamental chemical field. They appear most evidently in the case of open flows where irregularities of the field exhibit strong spatial intermittency as they are restricted to a fractal manifold. Nonuniformities of the Hölder exponent of the chemical field in closed flows appears as a consequence of the nonuniform stretching of the fluid elements. We study how this affects the scaling exponents of the structure functions, displaying anomalous scaling, and relate the scaling exponents to the distribution of local Lyapunov exponents of the advection dynamics. Theoretical predictions are compared with numerical experiments.