Construction of universal Thom-Whitney-a stratifications, their functoriality and Sard-type Theorem for singular varieties

{bf Construction.} For a dominating polynomial mapping {$F: K^nto K^l$} with an isolated critical value at 0 ($K$ an algebraically closed field of characteristic zero) we construct a closed {it bundle} $G_F subset T^{*}K^n $. We restrict $ G_F $ over the critical points $Sing(F)$ of $ F$ in $ F^{-1}(0)$ and partition $Sing(F)$ into {it quasistrata} of points with the fibers of $G_F$ of constant dimension. It turns out that T-W-a (Thom and Whitney-a) stratifications near $F^{-1}(0)$ exist iff the fibers of bundle $G_F$ are orthogonal to the tangent spaces at the smooth points of the quasistrata (e. g. when $ l=1$). Also, the latter are the orthogonal complements over an irreducible component $ S $ of a quasistratum only if $S $ is {bf universal} for the class of {T-W-a} stratifications, meaning that for any ${S_j}_j $ in the class, $ Sing (F) = cup_j S_j $, there is a component $S $ of an $ S_j $ with $Scap S$ being open and dense in both $S $ and $ S $. {bf Results.} We prove that T-W-a stratifications with only universal strata exist iff all fibers of $G_F$ are the orthogonal complements to the respective tangent spaces to the quasistrata, and then the partition of $Sing(F)$ by the latter yields the coarsest {it universal T-W-a stratification}. The key ingredient is our version of {bf Sard-type Theorem for singular spaces} in which a singular point is considered to be noncritical iff nonsingular points nearby are uniformly noncritical (e. g. for a dominating map $ F: X to Z $ meaning that the sum of the absolute values of the $ltimes l$ minors of the Jacobian matrix of $ F $, where $ l = dim (Z) $, not only does not vanish but, moreover, is separated from zero by a positive constant).