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{\bf Construction.} For a dominating polynomial mapping {$F: K^n\to K^l$}with an isolated critical value at 0 ($K$ an algebraically closed field ofcharacteristic 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 thefibers of $G_F$ of constant dimension. It turns out that T-W-a (Thom andWhitney-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 thequasistrata (e. g. when $ l=1$). Also, the latter are the orthogonalcomplements over an irreducible component $ S $ of a quasistratum only if $S $is {\bf universal} for the class of {T-W-a} stratifications, meaning that forany $\{S_j'\}_j $ in the class, $ \Sing (F) = \cup_j S'_j $, there is acomponent $S' $ of an $ S_j' $ with $S\cap S'$ being open and dense in both $S$ and $ S' $.  {\bf Results.} We prove that T-W-a stratifications with only universal strataexist iff all fibers of $G_F$ are the orthogonal complements to the respectivetangent spaces to the quasistrata, and then the partition of $\Sing(F)$ by thelatter yields the coarsest {\it universal T-W-a stratification}.  The key ingredient is our version of {\bf Sard-type Theorem for singularspaces} in which a singular point is considered to be noncritical iffnonsingular points nearby are 'uniformly noncritical' (e. g. for a dominatingmap $ F: X \to Z $ meaning that the sum of the absolute values of the $l\timesl$ minors of the Jacobian matrix of $ F $, where $ l = \dim (Z) $, not onlydoes not vanish but, moreover, is separated from zero by a positive constant).

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