  For a Brownian loop soup with intensity $c\in(0,1]$ in the unit disk, we show
that almost surely, the set of simple (resp. double) points on any portion of
boundary of any of its clusters has Hausdorff dimension $2-\xi_c(2)$ (resp.
$2-\xi_c(4)$), where $\xi_c(k)$ is the generalized disconnection exponent
computed in arxiv:1901.05436. As a consequence, when the dimension is positive,
such points are a.s. dense on every boundary of every cluster. There are a.s.
no triple points on the cluster boundaries.
  As an intermediate result, we establish a separation lemma for Brownian loop
soups, which is a powerful tool for obtaining sharp estimates on
non-intersection and non-disconnection probabilities in the setting of loop
soups. In particular, it allows us to define a family of generalized
intersection exponents $\xi_c(k, \lambda)$, and show that $\xi_c(k)$ is the
limit as $\lambda\searrow 0$ of $\xi_c(k, \lambda)$.
