A Riemannian foliation F can be given by local submersions on a transversal manifold T, the transverse coordinates transformations being local isometries of T. We study the differentiable equivalence class of the holonomy pseudogroup X of F ( which is the pseudogroup acting on T generated by the transverse coordinate transformations). When the foliated manifold is complete, X is a complete pseudogroup of isometries. Its closure for the c¹ topology is a Lie pseudogroup, and the orbit closures are embedded submanifolds of T.

We give a local model for X when the foliated manifold is simply connected. We then explain how these local models can be glued together when the space of leaf closures is of dimension less than or equal to 2. We thus obtain a classification of the holonomy pseudogroups of Riemannian foliations on complete simply connected spaces when the space of leaf closures is compact, of dimension less than or equal to 2.