  Given its well known spectral decomposition profile, the $1$-dim harmonic
oscillator potential modified by an inverse square ($1$-dim angular
momentum-like) contribution works as an efficient platform for probing
classical and quantum information quantifiers in the context of the phase-space
Weyl-Wigner formalism. In particular, the phase-space informational content
related to the canonical ensemble driven by such a {\em singular oscillator}
can be quantified in terms of well established analytical structures.
Considering that, on one hand, the {\em singular oscillator} produce a spectral
decomposition profile equivalent to that one of the unmodified harmonic system
-- in the sense that they result into identical thermodynamic statistics, even
for different statistical mixtures -- on the other hand, a more complete
scrutinization of their phase-space information content can capture some
different aspects of the encoded information for the related quantum ensembles.
Besides the identification of decoherence effects, the Wigner flow analysis is
presumedly useful in identifying stable quantum configurations, according to
finite temperature and interaction parameter values. Unexpectedly, our results
show that the equivalence between the statistical (quantum) mechanics of the
anharmonic singular oscillator and an ordinary harmonic oscillator can also be
extended to the phase-space quantum purity quantifier, which is analytically
computed and reproduces exactly the same quantum ensemble statistical mixture
profile, which does not depend on interaction parameter values.
