It is known that Laguerre-Gauss beams with indices n=0 and nonzero nz have a single phase singularity of order nz and the intensity shaped as a circumference. In this work a generalization of these beams is proposed, namely, for any closed curve on the plane there exists a family of singular beams dependind on a pair of integer-valued parameters, any member of which is structurally stable under propagation and focusing. In particular, when the curve is a circumference Laguerre-Gauss modes and parameters n,m are obtained.