We obtain sharp criteria for transverse stability and instability of line
solitons in the discrete nonlinear Schrödinger equations on one- and
two-dimensional lattices near the anti-continuum limit. On a two-dimensional
lattice, the fundamental line soliton is proved to be transversely stable
(unstable) when it bifurcates from the $X$ ($\Gamma$) point of the dispersion
surface. On a one-dimensional (stripe) lattice, the fundamental line soliton is
proved to be transversely unstable for both signs of transverse dispersion. If
this transverse dispersion has the opposite sign to the discrete dispersion,
the instability is caused by a resonance between isolated eigenvalues of
negative energy and the continuous spectrum of positive energy. These results
hold for both focusing and defocusing nonlinearities via a staggering
transformation. When the line soliton is transversely unstable, asymptotic
expressions for unstable eigenvalues are also derived. These analytical results
are compared with numerical results, and perfect agreement is obtained.