  In recent years, the study of the bienergy functional has attracted the
attention of a large community of researchers, but there are not many examples
where the second variation of this functional has been thoroughly studied. We
shall focus on this problem and, in particular, we shall compute the exact
index and nullity of some known examples of proper biharmonic maps. Moreover,
we shall analyse a case where the domain is not compact. More precisely, we
shall prove that a large family of proper biharmonic maps $\varphi:\mathbb{R}
\to \mathbb{S}^2$ is strictly stable with respect to compactly supported
variations. In general, the computations involved in this type of problems are
very long. For this reason, we shall also define and apply to specific examples
a suitable notion of index and nullity with respect to equivariant variations.
