In this paper, we are interested in the study of a shape optimization problem with Stokes constraints within the class of axisymmetric domains represented by the graph of a function. Existence results with weak assumptions on the regularity of the graph are provided. We strongly use these assumptions to get some topological properties. We formulate the (shape) optimization problem using different constraints formulations: uniform bound constraints on the function and its derivative and/or volume (global) constraint. Writing the first order optimality conditions allows us to provide quasi-explicit solutions in some particular cases and to give some hints for the treatment of the generic problem. Furthermore, we extend the (negative) result of [A. Henrot and Y. Privat, Arch. Ration. Mech. Anal., 196 (2010), pp. 281-302] dealing with the nonoptimality of the cylinder.