Most of the literature considers the standard model category structure on (graded) commutative differential algebras. But this generalizes to all (not-necessarily commutative) dg-algebras.

Details and references are listed here: nLab: model structure on dg-algebras.

What is known about how commutative dg-algebras sit inside all dg-algebras intrinsically ?

Do we have, for instance, a Quillen inclusion of the CommutativeDGAlgs into DGAlgs? Is its ajoint maybe a localization?