  In this paper, we investigate the instability of the spherical travelling
wave solutions for the Transport-Stokes system in $\mathbb{R}^3$. First, a
classical scaling argument ensures instability among all probability measures
for the Wasserstein metric and the $L^1$ norm. Secondly, we address the
instability among patch solutions with a perturbed surface. To this end, we
study the linearized system of a contour dynamics equation derived in [18] in
the case where the support of the patch is axisymmetric and described by
spherical parametrization. We investigate numerically the existence of positive
eigenvalues, which ensures the instability of the linearized system. Eventually
we recover numerically the instability of the travelling wave by solving the
Transport-Stokes equation using a finite element method on FreeFem.
