  Using the method of asymptotic splittings, the possible singularity
structures and the corresponding asymptotic behavior of a 3-brane in a
five-dimensional bulk are classified, in the case where the bulk field content
is parametrized by an analog of perfect fluid with an arbitrary equation of
state $P=\gamma\rho$ between the `pressure' $P$ and the `density' $\rho$. In
this analogy with homogeneous cosmologies, the time is replaced by the extra
coordinate transverse to the 3-brane, whose world-volume can have an arbitrary
constant curvature. The results depend crucially on the constant parameter
$\gamma$: (i) For $\gamma>-1/2$, the flat brane solution suffers from a
collapse singularity at finite distance, that disappears in the curved case.
(ii) For $\gamma<-1$, the singularity cannot be avoided and it becomes of the
type big rip for a flat brane. (iii) For $-1<\gamma\le -1/2$, the surprising
result is found that while the curved brane solution is singular, the flat
brane is not, opening the possibility for a revival of the self-tuning
proposal.
