  In this article, we revisit the question of fluctuations of linear statistics
of beta ensembles in the single cut and non-critical regime for general
potentials $V$ under mild regularity and growth assumptions. Our main objective
is to establish sharp quantitative Central Limit Theorems (CLT) for strong
distances, such as the total variation distance, which to the best of our
knowledge, is new for general potentials, even qualitatively. Namely, setting
$\mu_V$ the equilibrium measure, for a test function $\xi \in
\mathscr{C}^{14}$, we establish the convergence in total variation of
$X_n=\sum_{i=1}^n \xi(\lambda_i)-n\langle \xi,\mu_V\rangle$ to an explicit
Gaussian variable at the sharp speed $1/n$. Under the same assumptions, we also
establish multivariate CLTs for vectors of linear statistics in $p-$Wasserstein
distances for any $p\ge 1$, with the optimal rate $1/n$, a result which already
in dimension one sharpens the speed of convergence established in the recent
contribution [26] as well as the required regularity on the test functions. A
second objective of this paper, in a more qualitative direction, is to
establish the so-called super-convergence of linear statistics, that is to say
the convergence of all derivatives of the densities of $X_n$ uniformly on
$\mathbb{R}$, provided that $\xi\in\mathscr{C}^\infty(\mathbb{R})$ and is not
too degenerated in some sense.
