Coulomb gauge is quite peculiar for the behavior of the Green functions of a Yang-Mills theory. It makes computations more involved and recent lattice computations seem to point toward some different behavior with respect to the case of Landau gauge. Indeed, the propagator is dependent on the gauge choice and could seem not so much useful. We know that things do not stay this way from our understanding of simpler theories as quantum electrodynamics. Some physics can be extracted from them and, if analytically obtained, we can build up a quantum field theory. This explains the effort of a good part of the scientific community for their understanding in the low-energy limit of Yang-Mills theory.
While in the Landau gauge the gluon propagator on the lattice is seen to reach a finite non-null value at zero momenta, in the Coulomb gauge there are strong indications that the gluon propagator is strongly suppressed bending toward zero at lower momenta. This was firstly seen by Attilio Cucchieri (see here for a more recent analysis). This situation has been somehow improved in a recent paper (see here) in a Tokyo-Berlin-Adelaide collaboration. An understanding of the underlying physics would be improved through the analysis of the ghost propagator. This should be infrared enhanced. But from the paper of Cucchieri, Mendes and Maas that I cited above, it is not seen to change from a free behavior with changing the gauge. This is a relevant indicator and should be exploited wherever one tries to see the behavior of the propagators in different gauges.
Of course, the best way to have an insight on such Green functions is through analytical means. This is overdue in the case of the Coulomb gauge. Today on arxiv appeared a notable attempt by Alkofer, Maas and Zwanziger (see here). Alkofer and Zwanziger are pioneers in the use of Dyson-Schwinger equations for gauge theories and have given important contributions to quantum field theory in this area. So, their conclusions are relevant. They show that a more demanding truncation scheme is needed for this case while it appears not trivial at all the emerging of a linear rising potential. This paper puts the foundations for further work in this direction whose understanding is essential to Yang-Mills theory in the low-energy limit.