In this note, we use two case studies to analyze the relation between abnormal geodesiscs in time minimal control, accessibility and the regularity properties of the (time minimal) value function. The first problem concerns a Zermelo navigation problem in the plane. Using the historical problem of the calculus of variations set by Carathéodory and Zermelo aiming to compute the quickest nautical path of a ship on a river to be transferred from one shore to opposite one, we construct a semi-normal form to analyze the cusp singularity of the abnormal geodesic when meeting the transition between the strong and weak current domains and to evaluate the time minimal value function. The problem is related to reaching the boundary of the two domains in minimum time. The second problem concerns the classification of the time minimal syntheses for chemical reactors near a terminal manifold of codimension one, in particular in relation with the McKeithan network and associated to maximize the production of one chemical species. Both cases studies have the common geometric frame and computations techniques illustrate the role of singularity theory in geometric optimal control, in relation with abnormal geodesics and regularity of the value function solution of Hamilton-Jacobi-Bellman equation.