In this paper, we analyse a model for the growth of three-dimensional walled cells. In this model the biomechanical expansion of the cell is coupled with the geometry of its wall. We consider that the density of building material depends on the curvature of the cell wall, thus yield-ing possible anisotropic growth. The dynamics of the axisymmetric cell wall is described by a system of nonlinear PDE including a nonlin-ear convection-diffusion equation coupled with a Poisson equation. We develop the linear stability analysis of the spherical symmetric config-uration in expansion. We identify three critical parameters that play a role in the possible instability of the radially symmetric shape, namely the degree of nonlinearity of the coupling, the effective diffusion of the building material, and the Poisson's ratio of the cell wall. We also investigate numerically pattern selection in the nonlinear regime. All the results are also obtained for a simpler, but similar, two-dimensional model.