  We investigate the zero-temperature phase diagram of the fully frustrated
transverse field Ising model on the square lattice both in the classical limit
and in the presence of quantum fluctuations. At the classical level (the limit
of infinite spin $S$), we find that upon decreasing the transverse field
$\Gamma$ this model exhibits a phase transition from the fully polarized state
into an eight-fold degenerate translational symmetry breaking state. This phase
can be identified to correspond to plaquette order in the dimer language and
remains the lowest-energy state in the entire range of fields below the
critical one, $\Gamma_c$. The eight-fold degenerate solution which corresponds
to columnar order in the dimer language is a saddle point of the classical
energy. It is degenerate with the plaquette solution at $\Gamma=0$ and is only
slightly higher in energy in the whole interval $0<\Gamma<\Gamma_c$. The effect
of quantum fluctuations is investigated in the context of a large S expansion
both for the plaquette and columnar structures. For this purpose we employ an
approximate method allowing to estimate from above the fluctuation-induced
correction to the energy of a configuration which at the classical level is a
saddle point of the energy, \textit{not} a local minimum. Although the
convergence of the $1/S$ expansion in the $\Gamma/J\rightarrow 0$ limit remains
an open question, harmonic quantum fluctuations show a clear tendency to
overcome the energy difference between the two states and to change the
classical picture favoring the columnar order over the plaquette one in a wide
parameter range.
