  Indices of singular points of a vector field or of a 1-form on a smooth
manifold are closely related with the Euler characteristic through the
classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive
topological invariants of spaces with some additional structures) are sometimes
related with corresponding analogues of indices of singular points. Earlier
there was defined a notion of the universal Euler characteristic of an
orbifold. It takes values in a ring R, as an abelian group freely generated by
the generators, corresponding to the isomorphism classes of finite groups. Here
we define the universal index of an isolated singular point of a vector field
or of a 1-form on an orbifold as an element of the ring R. For this index, an
analogue of the Poincar\'e-Hopf theorem holds.
