The critical behavior of a three real parameter class of solutions of the sixth Painlev ́e
equation is computed, and parametrized in terms of monodromy data of the associated
2 × 2 matrix linear Fuchsian system of ODE. The class may contain solutions with poles
accumulating at the critical point. The study of this class closes a gap in the description
of the transcendents in an one to one correspondence with the monodromy data. These
transcendents are reviewed in the paper. Some formulas that relate the monodromy data
to the critical behaviors of the four real (two complex) parameter class of solutions
are missing in the literature, so they are computed here. A computational procedure
to write the full expansion of the four and three real parameter class of solutions is
proposed.
