  We prove strong nonlinear illposedness results for the generalized SQG
equation $$\partial_t \theta + \nabla^\perp \Gamma[\theta] \cdot \nabla \theta
= 0 $$ in any sufficiently regular Sobolev spaces, when $\Gamma$ is a singular
in the sense that its symbol satisfies $|\Gamma(\xi)|\to\infty$ as
$|\xi|\to\infty$ with some mild regularity assumptions. The key mechanism is
degenerate dispersion, i.e., the rapid growth of frequencies of solutions
around certain shear states, and the robustness of our method allows one to
extend linear and nonlinear illposedness to fractionally dissipative systems,
as long as the order of dissipation is lower than that of $\Gamma$. Our
illposedness results are completely sharp in view of various existing
wellposedness statements as well as those from our companion paper. Key to our
proofs is a novel construction of degenerating wave packets for the class of
linear equations $$\partial_t \phi + ip(t,X,D)\phi = 0$$ where $p(t,X,D)$ is a
pseudo-differential operator which is self-adjoint in $L^2$, degenerate, and
dispersive. Degenerating wave packets are approximate solutions to the above
linear equation with spatial and frequency support localized at
$(X(t),\Xi(t))$, which are solutions to the bicharacteristic ODE system
associated with $p(t,x,\xi)$. These wave packets explicitly show degeneration
as $X(t)$ approaches a point where $p$ vanishes, which in particular allows us
to prove illposedness in topologies finer than $L^2$. While the equation for
the wave packet can be formally obtained from a Taylor expansion of the symbol
near $\xi=\Xi(t)$, the difficult part is to rigorously control the error in
sufficiently long timescales, which is obtained by sharp estimates for not only
degenerating wave packets but also for oscillatory integrals which naturally
appear in the error estimate.
