  We review studies on the singularity structure and asymptotic analysis of a
3-brane (flat or curved) embedded in a five-dimensional bulk filled with a
`perfect fluid' with an equation of state with the `pressure' and the `density'
of the fluid depending on the fifth space coordinate. Regular solutions
satisfying positive energy conditions in the bulk exist only in the cases of a
flat brane with an EoS parameter equal to -1, or of AdS branes for EoS
parameter values in suitable intervals. More cases can be found by gluing two
regular branches of solutions at the position of the brane. However, only the
case of a flat brane with an EoS parameter equal to -1 leads to finite Planck
mass on the brane and thus localises gravity. In a more recent work, we showed
that a way to rectify the previous findings and obtain a solution for a flat
brane in a finite range of the EoS parameter, which is both free from
finite-distance singularities and compatible with the physical conditions of
energy and finiteness of four-dimensional Planck mass, is by introducing a bulk
fluid component that satisfies a nonlinear equation of state.
