  We study various triangulated motivic categories and introduce a vast family
of aisles (these are certain classes of objects) in them. These aisles are
defined in terms of the corresponding "motives" (or motivic spectra) of smooth
varieties in them; we relate them to the corresponding homotopy t-structures.
We describe our aisles in terms of stalks at function fields and prove that
they widely generalize the ones corresponding to slice filtrations. Further,
the filtrations on the "homotopy hearts" $Ht_{hom}^{eff}$ of the corresponding
effective subcategories that are induced by these aisles can be described in
terms of (Nisnevich) sheaf cohomology as well as in terms of the Voevodsky
contractions $-_{-1}$. Respectively, we express the condition for an object of
$Ht_{hom}^{eff}$ to be weakly birational (i.e., that its $n+1$th contraction is
trivial or, equivalently, the Nisnevich cohomology vanishes in degrees $>n$ for
some $n\ge 0$) in terms of these aisles; this statement generalizes well-known
results of Kahn and Sujatha. Next, these classes define weight structures
$w_{Smooth}^{s}$ (where $s=(s_{j})$ are non-decreasing sequences parameterizing
our aisles) that vastly generalize the Chow weight structures $w_{Chow}$
defined earlier. Using general abstract nonsense we also construct the
corresponding adjacent $t-$structures $t_{Smooth}^{s}$ and prove that they give
the birationality filtrations on $Ht^{eff}_{hom}$.
  Moreover, some of these weight structures induce weight structures on the
corresponding $n-$birational motivic categories (these are the localizations by
the levels of the slice filtrations). Our results also yield some new
unramified cohomology calculations.
