  We study singular radially symmetric solution of the stationary Keller-Segel
equation, that is, an elliptic equation with exponential nonlinearity, which is
super-critical in dimension $N \geq 3$. The solutions are unbounded at the
origin and we show that they describe the asymptotics of bifurcation branches
of regular solutions. It is shown that for any ball and any $k \geq 0$, there
is a singular solution that satisfies Neumann boundary condition and oscillates
at least $k$ times around the constant equilibrium. Moreover, we prove that in
dimension $3 \leq N \leq 9$ there are regular solutions satisfying Neumann
boundary conditions that are close to singular ones. Hence, it follows that
there exist regular solutions on any ball with arbitrarily fast oscillations.
For generic radii, we show that the bifurcation branches of regular solutions
oscillate in the bifurcation plane when $3\leq N\leq 9$ and approach to a
singular solution. In dimension $N > 10$, we show that the Morse index of the
singular solution is finite, and therefore the existence of regular solutions
with fast oscillations is not expected.
