Cauchy's singular integral operator and its beautiful spectrum

The Cauchy singular integral operator, S, is one of the main actors in the theory of Fourier convolutions, Toeplitz operators, Riemann-Hilbert problems, Wiener-Hopf and singular integral equations. While the boundedness of S has been studied for many decades, final results on the spectrum of S were obtained only in recent times. During the last few years, it was discovered that there is a surprising and undreamt-of metamorphosis of the (local) spectra of S from circular arcs via horns and logarithmic double-spirals to so-called logarithmic leaves with a halo. This article is a survey of this fascinating development. We hope it is both a feast for the eyes and an illustration of Nick Trefethen's motto: The spectrum gives an operator a personality!