  We study spectral properties of the Neumann-Poincar\'e operator on planar
domains with corners with particular emphasis on existence of continuous
spectrum and pure point spectrum. We show that the rate of resonance at
continuous spectrum is different from that at eigenvalues, and then derive a
method to distinguish continuous spectrum from eigenvalues. We perform
computational experiments using the method to see whether continuous spectrum
and pure point spectrum appear on domains with corners. For the computations we
use a modification of the Nystr\"om method which makes it possible to construct
high-order convergent discretizations of the Neumann-Poincar\'e operator on
domains with corners. The results of experiments show that all three possible
spectra, absolutely continuous spectrum, singularly continuous spectrum, and
pure point spectrum, may appear depending on domains. We also prove rigorously
two properties of spectrum which are suggested by numerical experiments:
symmetry of spectrum (including continuous spectrum), and existence of
eigenvalues on rectangles of high aspect ratio.
