We propose a novel class of refractive optical elements by wrinkling the conical surface of a usual (conical) axicon, which leads to geometrical singularities (cusps). Such wrinkled axicons have been fabricated at the micron scale by using three-dimensional femtosecond-laser photopolymerization technique and we report on their experimental and numerical characterization. The beam shaping capabilities of these structures are discussed for both intensity and phase, which includes topological beam shaping that results from azimuthally modulated optical spin-orbit interaction.