The continuous spectrum of an operator may not look like an interval at all. In fact, it can very well be the Cantor set!
Here is an example:
Let $\mu $ be any positive Borel measure on the Cantor set, here denoted by $C$, such that $\mu $ has no atoms
(that is, $\mu (\{x\})=0$, for all $x$ in $C$) and full support
(that is, $\mu (U)>0$, for all nonempty
open sets $U\subseteq C$).
See below for the construction of such a measure.
Consider the operator $T$ on $L^2(C,\mu )$ given by
$$
T(\xi )|_x = h(x)\xi (x), \quad \forall \xi \in L^2(C,\mu ), \quad \forall x\in C,
$$
where $h$ is the function defined on $C$ by $h(x)=x$.
Using this wikipedia page it follows
that the spectrum of $T$ coincides with its continuous spectrum, which in turn coincides with the range of $h$, a.k.a $C$.
Here is one way to construct a measure $\mu $ on the Cantor set with the required properties. First of all recall that
the Cantor set is homeomorphic to $\{0,1\}^{\mathbb N}$, also known as Bernouli's space. For a concrete homeomorphism
take
$
\varphi :\{0,1\}^{\mathbb N}\to C,
$
given by
$$
\varphi (x) = \sum_{n=1}^\infty x_n3^{-n},
$$
for every $x=(x_1,x_2,\ldots ) \in \{0,1\}^{\mathbb N}$.
Consider the uniform probability measure $\rho $ on $\{0,1\}$, given by $\rho (\{0\}) =\rho (\{1\}) =1/2$, and let
$$
\nu =\prod_{n=1}^\infty \rho
$$
be the corresponding product measure. Incidentally $\nu $ is known as the Bernouli measure.
It is well known (and easy to prove) that $\nu $ has no atoms and full support. Since $\varphi $ is a homeomorphism, it
follows that the push forward measure $\mu := \varphi _*(\nu )$ has the same properties.
Best Answer
I myself am not aware of a dynamical characterization of absolutely or singular continuous spectra with the same generality as the RAGE theorem.
If you are willing to restrict the scope to quantum mechanics and the theory of Schrödinger operators, however, you get sort of a dynamical characterization of absolutely continuous spectrum: establishing absence of singular continuous spectrum is related to asymptotic completeness, i.e. the density of scattering states plus bound states. See e.g. Reed, Simon, Vol. 4, Sec. XIII.6.
This is of course not on the same level of generality as RAGE, because you require a free time evolution to compare to.