  The formation of singularities in solutions to the dispersionless
Kadomtsev-Petviashvili (dKP) equation is studied numerically for different
classes of initial data. The asymptotic behavior of the Fourier coefficients is
used to quantitatively identify the critical time and location and the type of
the singularity. The approach is first tested in detail in 1+1 dimensions for
the known case of the Hopf equation, where it is shown that the break-up of the
solution can be identified with prescribed accuracy. For dissipative
regularizations of this shock formation as the Burgers' equation and for
dispersive regularizations as the Korteweg-de Vries equation, the Fourier
coefficients indicate as expected global regularity of the solutions. The
Kadomtsev-Petviashvili (KP) equation can be seen as a dispersive regularization
of the dKP equation. The behavior of KP solutions for small dispersion
parameter $\epsilon\ll 1$ near a break-up of corresponding dKP solutions is
studied. It is found that the difference between KP and dKP solutions for the
same initial data at the critical point scales roughly as $\epsilon^{2/7}$ as
for the Korteweg-de Vries equation.
