As the discussion here Is singular cohomology representable by a (Voevodsky's) motivic complex? shows, the singular cohomology of (smooth) complex varieties is represented by a motivic complex (and also by a motivic spectrum). My question is: what can be proved about the ring structure for this complex/spectrum? Is it known that this a 'weak' ring spectrum? an $A_{\infty}$spectrum? a highly structured ring spectrum? Any hints would be very welcome!
Singular cohomology is represented by an $E_\infty$ motivic ring spectrum. That spectrum is $\mathbf{R}f(H\mathbb{Z})$ where $f$ is right adjoint to the stable topological realization functor and $H\mathbb{Z}$ is the topological EilenbergMac Lane spectrum. Since this is a symmetric monoidal Quillen adjunction (see Theorem A.45 here: http://arxiv.org/pdf/0709.3905.pdf), $\mathbf{R}f$ preserves $E_\infty$objects. The same argument should show that the motivic complex is also $E_\infty$, but I don't know a reference for the required symmetric monoidal adjunction. At least if you work with $(\infty,1)$categories this adjunction comes for free from the fact that motivic complexes are the same thing as modules over the motivic EilenbergMac Lane spectrum, whose topological realization is $H\mathbb{Z}$. 

