There are no singularities in any bizarre sense. There are singularities on hairy balls, but they’re just cowlicks. There are singularities in Falaco solitons, but they’re just centres of rotation. A Falaco soliton is like half a smoke ring. Smoke rings were the subject of experiments by Thomson and Tait, the vortex atom guys who coined the phrase spherical harmonics. They were ahead of their time, because the electron is an optical vortex. There’s a centre of rotation in the electron too: