  We define an $L^2$-signature for proper actions on spaces of leaves of
transversely oriented foliations with bounded geometry. This is achieved by
using the Connes fibration to reduce the problem to the case of Riemannian
bifoliations where we show that any transversely elliptic first order operator
in an appropriate Beals-Greiner calculus, satisfying the usual axioms, gives
rise to a semi-finite spectral triple over the crossed product algebra of the
foliation by the action, and hence a periodic cyclic cohomology class through
the Connes-Chern character. The Connes-Moscovici hypoelliptic signature
operator yields an example of such a triple and gives the differential
definition of our "$L^2$-signature". For Galois coverings of bounded geometry
foliations, we also define an Atiyah-Connes semi-finite spectral triple which
generalizes to Riemannian bifoliations the Atiyah approach to the $L^2$-index
theorem. The compatibility of the two spectral triples with respect to Morita
equivalence is proven, and by using an Atiyah-type theorem proven in [BH17b],
we deduce some integrality results for Riemannian foliations with torsion-free
monodromy groupoids.
