We consider a mathematical model of a spherical tumor within a tissue whose material properties are either similar to a porous medium, or to a fluid. When the proliferation rate increases and the cell-to-cell adhesiveness decreases, the tumor will become unstable, and its spherical shape will change, producing several protrusions, say N. We prove that whereas for a tumor in a porous medium N is just 2, for a tumor in fluid tissue N may be arbitrarily large depending on the size of the original tumor’s radius. Mathematically, the model deals with a bifurcation problem for a system of PDEs with a free boundary.