  We introduce the notion of equivariant basic cohomology for singular
Riemannian foliations with transverse infinitesimal actions, and prove some
elementary properties such as its invariance under homotopies. For the
particular case of singular Killing foliations, there is the natural transverse
action of its structural algebra. We prove that, modulo torsion, its
equivariant basic cohomology localizes to the set of closed leaves of the
foliation, in the spirit of the classical localization theorem of Borel. As
applications, we obtain that the basic Euler characteristic also localizes to
this set, and that the dimension of the basic cohomology of the localized
foliation is less than or equal to that of the whole foliation, with equality
occurring precisely in the equivariantly formal case.
