  In this paper, we initiate the study of the instability of naked
singularities without symmetries. In a series of papers, Christodoulou proved
that naked singularities are not stable in the context of the spherically
symmetric Einstein equations coupled with a massless scalar field. We study in
this paper the next simplest case: a characteristic initial value problem of
this coupled system with the initial data given on two intersecting null cones,
the incoming one of which is assumed to be spherically symmetric and singular
at its vertex, and the outgoing one of which has no symmetries. It is shown
that, arbitrarily fixing the initial scalar field, the set of the initial
conformal metrics on the outgoing null cone such that the maximal future
development does not have any sequences of closed trapped surfaces approaching
the singularity, is of first category in the whole space in which the shear
tensors are continuous. Such a set can then be viewed as exceptional, although
the exceptionality is weaker than the at least $1$ co-dimensionality in
spherical symmetry. Almost equivalently, it is also proved that, arbitrarily
fixing an incoming null cone $\underline{C}_\varepsilon$ to the future of the
initial incoming null cone, the set of the initial conformal metrics such that
the maximal future development has at least one closed trapped surface before
$\underline{C}_\varepsilon$, contains an open and dense subset of the whole
space. Since the initial scalar field can be chosen such that the singularity
is naked if the initial shear is set to be zero, we may say that the spherical
naked singularities of a self-gravitating scalar field are not stable under
gravitational perturbations. This in particular gives new families of
non-spherically symmetric gravitational perturbations different from the
original spherically symmetric scalar perturbations given by Christodoulou.
