  We present new results involving general higher-curvature gravities in three
dimensions. The most general Lagrangian can be written as a function of the
Ricci scalar $R$, $\mathcal{S}_2\equiv \tilde R_{a}^b \tilde R_b^a$ and
$\mathcal{S}_3\equiv \tilde R_a^b \tilde R_b^c \tilde R_c^a$ where $\tilde
R_{ab}$ is the traceless part of the Ricci tensor. First, we provide a formula
for the exact number of independent order-$n$ densities, $\#(n)$. This
satisfies the identity $\#(n-6)=\#(n)-n$. Then, we show that, linearized around
a general Einstein solution, a generic order-$n\geq 2$ density can be written
as a linear combination of $R^n$, which does not propagate the generic massive
graviton, plus a density which does not propagate the generic scalar mode,
$R^n-12n(n-1)R^{n-2}\mathcal{S}_2$, plus $\#(n)-2$ densities which contribute
trivially to the linearized equations. Next, we obtain an analytic formula for
the quasinormal frequencies of the BTZ black hole for a general theory. Then,
we provide a recursive formula as well as a general closed expression for
order-$n$ densities which non-trivially satisfy an holographic c-theorem,
clarify their relation with Born-Infeld gravities and prove that they never
propagate the scalar mode. We show that at each order there exist $\#(n-6)$
densities which satisfy the holographic c-theorem trivially and that all of
them are proportional to a single sextic density $\Omega_{(6)}\equiv 6
\mathcal{S}_3^2-\mathcal{S}_2^3$. We prove that there are also $\#(n-6)$
order-$n$ Generalized Quasi-topological densities in three dimensions, all of
which are "trivial" in the sense of making no contribution to the metric
function equation. The set of such densities turns out to coincide exactly with
the one of theories trivially satisfying the holographic c-theorem. We comment
on the relation of $\Omega_{(6)}$ to the Segre classification of
three-dimensional metrics.
