In the context of analysis of spectra of perturbed harmonic oscillator we consider basic questions of spectral analysis: convergence (or divergence) of spectral decompositions, structure of zones of instability, asymptotic of eigenvalues, growth of the norms of spectral projections. Special attention is given to the case of perturbation by two-point odd delta-potential. Significant part of discussion is based on my joint works with J.Adduci, P.Djakov, P.Siegl, J.Viola.