To quote from Will's book (Theory and Experiment in Gravitational Physics, Rev. Ed., Cambridge, 1993), "[...] in almost all experiments discussed in this book, the observable effects of torsion are negligible".
Will then mentions a counterexample (Ni, Phys. Rev. D 19, 2260 (1979)), but that example is a specific theory in which torsion propagates and couples to the electromagnetic field.
In another reference cited by Will (Hehl et al., Rev. Mod. Phys. 48, 393 (1976)), which is a review of spin and torsion, the authors explain that torsion normally does not propagate, and that therefore "there can be no torsion of space-time outside the spinning matter distribution itself. Torsion is inextricably bound to matter and cannot propagate through the vacuum as a torsion wave or via any interaction of nonvanishing range" (emphasis is the authors').
In short, Will then concludes that although torsion is one of the subjects that "could generate a monograph of its own", its effects are indeed negligible, and other than providing the aforementioned references, he then ignores the topic in the rest of his book. In particular, torsion plays no role in the parameterized post-Newtonian (PPN) expansion employed in the monograph.
I ran into this "nondynamical" nature of torsion myself when I was exploring an idea related to Moffat's STVG (Scalar-Tensor-Vector Gravity) modified gravity theory.