Unconstrained action in Yang-Mills theory Is there any gauge in which Yang-Mills theory (4d, non-SUSY) can be written as a local action containing only the propagating modes?
 A: 1) The usual definition of a local action forbids the Lagrangian density ${\cal L}(x)$ to contain inverse differential operators (=integration operators) like  $\partial_\mu^{-1}$, which is a non-local operation. 
If OP requires the action to be on manifestly local form in the above sense, then the answer is No, to the best of my knowledge.
2) On the other hand, if one does not require a manifestly local form, then it is possible to  write the gauge-fixed Yang-Mills action in $d\geq 2$ spacetime dimensions using propagating transversal gauge potentials only. 
Consider the light-cone formulation (which implies that evolution/propagation is wrt. the light-cone coordinate $x^+$ as opposed to $x^0\equiv t$) in the light-cone gauge $A^+=0$. 
Then the gauge potential $A^{-}$ becomes a non-propagating auxiliary field, which can be integrated out. This yields a Lagrangian density that only contains $d-2$ propagating transversal fields, but with inverse powers of $\partial_-$, and hence non-local. 
See e.g. Warren Siegel, Fields, p. 210, arXiv:hep-th/9912205, for further details.
