  We explore the concept of eigenvalue avoidance, which is well understood for
real symmetric and Hermitian matrices, for other classes of structured
matrices. We adopt a differential geometric perspective and study the generic
behaviour of the eigenvalues of regular and injective curves $t \in ]a,b[
\mapsto A(t) \in \mathcal{N} $ where $\mathcal{N}$ is a smooth real Riemannian
submanifold of either $\mathbb{R}^{n \times n}$ or $\mathbb{C}^{n \times n}$.
We focus on the case where $\mathcal{N}$ corresponds to some class of (real or
complex) structured matrices including skew-symmetric, skew-Hermitian,
orthogonal, unitary, banded symmetric, banded Hermitian, banded skew-symmetric,
and banded skew-Hermitian. We argue that for some structures eigenvalue
avoidance always happens, whereas for other structures this may depend on the
parity of the size, on the numerical value of the multiple eigenvalue, and
possibly on the value of determinant. As a further application of our tools we
also study singular value avoidance for unstructured matrices in $\mathbb{R}^{m
\times n}$ or $\mathbb{C}^{m \times n}$.
