We shall study the growth of a bacterial colony on an agar substrate, modeled by reaction-diffusion equations for the concentration of bacteria and nutrients, with the more realistic bacterial diffusivity dependent on their concentration. This dependence can give the bacterial concentration front a
(a) finite width and
(b) constant asymptotic speed, but also a
(c) singular character. (a,b) enable an effective moving-boundary description of the edge of the colony, which we propose to derive, while (c) complicates it, but the problem can be solved, uncovering special properties of singular fronts. The resulting moving-boundary model will reveal the differences and similarities with known free-boundary problems. Since it will not correspond exactly to any of them, we will study its linear regime analytically, that of the reaction-diffusion model analytically and numerically, and compare them, in order to elucidate the effect of the model parameters on the thickness of the front and the development of the instability. This will then guide a study of the nonlinear dynamics (weakly nonlinear analysis of the free-boundary problem and/or simulations of the reaction-diffusion model), which will help extend the conclusions to the final morphology diagram. The extension of the results to other problems with on-constant diffusivity will also be addressed with special care.