  We examine the slice spectral sequence for the cohomology of singular schemes
with respect to various motivic T-spectra, especially the motivic cobordism
spectrum. When the base field k admits resolution of singularities and X is a
scheme of finite type over k, we show that Voevodsky's slice filtration leads
to a spectral sequence for MGL(X) whose terms are the motivic cohomology groups
of X defined using the cdh-hypercohomology. As a consequence, we establish an
isomorphism between certain geometric parts of the motivic cobordism and
motivic cohomology of X. A similar spectral sequence for the connective
K-theory leads to a cycle class map from the motivic cohomology to the homotopy
invariant K-theory of X. We show that this cycle class map is injective for
projective schemes. We also deduce applications to the torsion in the motivic
cohomology of singular schemes.
