  We present a systematic study of a new type of consistent ``Brane-world
Kaluza-Klein Reduction,'' which describe fully non-linear deformations of
co-dimension one objects that arise as solutions of a large class of gauged
supergravity theories in diverse dimensions, and whose world-volume theories
are described by ungauged supergravities with one half of the original
supersymmetry. In addition, we provide oxidations of these Ansatze which are in
general related to sphere compactified higher dimensional string theory or
M-theory. Within each class we also provide explicit solutions of brane
configurations localised on the world-brane. We show that at the Cauchy horizon
(in the transverse dimension of the consistently Kaluza-Klein reduced
world-brane) there is a curvature singularity for any configuration with a
non-null Riemann curvature or a non-vanishing Ricci scalar that lives in the
world-brane. Since the massive Kaluza-Klein modes can be consistently
decoupled, they cannot participate in regulating these singularities.
