  We develop stability analysis for matter-wave solitons in a two-dimensional
(2D) Bose-Einstein condensate loaded in an optical lattice (OL), to which
periodic time modulation is applied, in different forms. The stability is
studied by dint of the variational approximation and systematic simulations.
For solitons in the semi-infinite gap, well-defined stability patterns are
produced under the action of the attractive nonlinearity, clearly exhibiting
the presence of resonance frequencies. The analysis is reported for several
time-modulation formats, including the case of in-phase modulations of both
quasi-1D sublattices, which build the 2D square-shaped OL, and setups with
asynchronous modulation of the sublattices. In particular, when the modulations
of two sublattices are phase-shifted by {\delta}={\pi}/2, the stability map is
not improved, as the originally well-structured stability pattern becomes fuzzy
and the stability at high modulation frequencies is considerably reduced. Mixed
results are obtained for anti-phase modulations of the sublattices
({\delta}={\pi}), where extended stability regions are found for low modulation
frequencies, but for high frequencies the stability is weakened. The analysis
is also performed in the case of the repulsive nonlinearity, for solitons in
the first finite bandgap. It is concluded that, even though stability regions
may be found, distinct stability boundaries for the gap solitons cannot be
identified clearly. Finally, the stability is also explored for vortex solitons
of both the "square-shaped" and "rhombic" types (i.e., off- and
on-site-centered ones).
