  Finite Sample Smeariness (FSS) has been recently discovered. It means that
the distribution of sample Fr\'echet means of underlying rather unsuspicious
random variables can behave as if it were smeary for quite large regimes of
finite sample sizes. In effect classical quantile-based statistical testing
procedures do not preserve nominal size, they reject too often under the null
hypothesis. Suitably designed bootstrap tests, however, amend for FSS. On the
circle it has been known that arbitrarily sized FSS is possible, and that all
distributions with a nonvanishing density feature FSS. These results are
extended to spheres of arbitrary dimension. In particular all rotationally
symmetric distributions, not necessarily supported on the entire sphere feature
FSS of Type I. While on the circle there is also FSS of Type II it is
conjectured that this is not possible on higher-dimensional spheres.
