  Spectrahedra are affine-linear sections of the cone $\mathcal{P}_n$ of
positive semidefinite symmetric $n\times n$-matrices. We consider random
spectrahedra that are obtained by intersecting~$\mathcal{P}_n$ with the
affine-linear space $\mathbf{1} + V$, where $\mathbf{1}$ is the identity matrix
and $V$ is an $\ell$-dimensional linear space that is chosen from the unique
orthogonally invariant probability measure on the Grassmanian of $\ell$-planes
in the space of $n\times n$ real symmetric matrices (endowed with the Frobenius
inner product). Motivated by applications, for $\ell=3$ we relate the average
number $\mathbb{E} \sigma_n$ of singular points on the boundary of a
three-dimensional spectrahedron to the volume of the set of symmetric matrices
whose two smallest eigenvalues coincide. In the case of quartic spectrahedra
($n=4$) we show that $\mathbb{E} \sigma_4 = 6-\frac{4}{\sqrt{3}}$. Moreover, we
prove that the average number $\mathbb{E} \rho_n$ of singular points on the
real variety of singular matrices in $\mathbf{1} + V$ is $n(n-1)$. This
quantity is related to the volume of the variety of real symmetric matrices
with repeated eigenvalues. Furthermore, we compute the asymptotics of the
volume and the volume of the boundary of a random spectrahedron.
