  In this paper, we study existence, regularity, classification, and
asymptotical behaviors of solutions of some Monge-Amp\`ere equations with
isolated and line singularities. We classify all solutions of $\det \nabla^2
u=1$ in $\R^n$ with one puncture point. This can be applied to characterize
ellipsoids, in the same spirit of Serrin's overdetermined problem for the
Laplace operator. In the case of having $k$ non-removable singular points for
$k>1$, modulo affine equivalence the set of all generalized solutions can be
identified as an explicit orbifold of finite dimension. We also establish
existence of global solutions with general singular sets, regularity
properties, and optimal estimates of the second order derivatives of
generalized solutions near the singularity consisting of a point or a straight
line. The geometric motivation comes from singular semi-flat Calabi-Yau
metrics.
