  Given a manifold M with a submanifold N, the deformation space D(M,N) is a
manifold with a submersion to R whose zero fiber is the normal bundle, and all
other fibers are equal to M. This article uses deformation spaces to study the
local behavior of various geometric structures associated with singular
foliations, with N a submanifold transverse to the foliation. New examples
include L_\infty-algebroids, Courant algebroids, and Lie bialgebroids. In each
case, we obtain a normal form theorem around N, in terms of a model structure
over the normal bundle.
