  We introduce a complete obstruction to the existence of nonvanishing vector
fields on a closed orbifold $Q$. Motivated by the inertia orbifold, the space
of multi-sectors, and the generalized orbifold Euler characteristics, we
construct for each finitely generated group $\Gamma$ an orbifold called the
space of $\Gamma$-sectors of $Q$. The obstruction occurs as the Euler-Satake
characteristics of the $\Gamma$-sectors for an appropriate choice of $\Gamma$;
in the case that $Q$ is oriented, this obstruction is expressed as a cohomology
class, the $\Gamma$-Euler-Satake class. We also acquire a complete obstruction
in the case that $Q$ is compact with boundary and in the case that $Q$ is an
open suborbifold of a closed orbifold.
