  We investigate equivariant and invariant topological complexity of spheres
endowed with smooth non-free actions of cyclic groups of prime order. We prove
that semilinear $\mathbb{Z}/p$-spheres have both invariants either $2$ or $3$
and calculate exact values in all but two cases for linear actions. On the
other hand, we exhibit examples which show that these invariants can be
arbitrarily high in the class of smooth $\mathbb{Z}/p$-spheres.
