  We investigate both linear and nonlinear stability aspects of rigid motions
(resp. M\"obius transformations) of $\mathbb{S}^{n-1}$ among Sobolev maps from
$\mathbb{S}^{n-1}$ into $\mathbb{R}^n$. Unlike similar in flavour results for
maps defined on domains of $\mathbb{R}^n$ and mapping into $\mathbb{R}^n$, not
only an isometric (resp. conformal) deficit is necessary in this more flexible
setting, but also a deficit measuring the distortion of $\mathbb{S}^{n-1}$
under the maps in consideration. The latter is defined as an associated
isoperimetric type of deficit. We mostly focus on the case $n=3$, where we also
explain why the estimates are optimal in their corresponding settings. In the
isometric case the estimate holds true also when $n=2$ and generalizes in
dimensions $n\geq 4$ as well, if one requires apriori boundedness in a certain
higher Sobolev norm. We also obtain linear stability estimates for both cases
in all dimensions. These can be regarded as Korn-type inequalities for the
combination of the quadratic form associated with the isometric (resp.
conformal) deficit on $\mathbb{S}^{n-1}$ and the isoperimetric one.
