  We study singular radially symmetric solution to the Lin-Ni-Takagi equation
for a supercritical power non-linearity in dimension $N\geq 3$. 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 show that the Morse index of the
singular solution is finite or infinite if the exponent is respectively larger
or smaller than the Joseph--Lundgren exponent.
