  We prove that the asymptotic distribution of resonances has a multilevel
internal structure for the following classes of Hamiltonians H: Schr\"odinger
operators with point interactions in $\mathbb{R}^3$, quantum graphs, and 1-D
photonic crystals. In the case of $N \ge 2$ point interactions, the set of
resonances $\Sigma (H)$ essentially consists of a finite number of sequences
with logarithmic asymptotics. We show how the leading parameters $\mu$ of these
sequences are connected with the geometry of the set $Y=\{y_j\}_{j=1}^N$ of
interaction centers $y_j \in \mathbb{R}^3$. The minimal parameter $\mu^{min}$
corresponds to the sequences with `more narrow' and so more observable
resonances. The asymptotic density of such narrow resonances is described by
the multiplicity of $\mu^{\min}$, which occurs to be connected with the
symmetries of Y and naturally introduces a finite number of classes of
configurations of $Y$. In the case of quantum graphs and 1-D photonic crystals,
the decomposition of $\Sigma(H)$ into a finite number of asymptotic sequences
is proved under additional commensurability conditions. To address the case of
a general quantum graph, we introduce families of special counting and
asymptotic density functions for two types of curved complex strips. The
obtained results and effects are compared with those of obstacle scattering.
