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 ν(M, N), 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∞-algebroids, Courant algebroids, and Lie bialgebroids. In each case, we obtain a normal form theor...