The Euler class of the normal bundle of a Seifert fibration and horizontal foliations

For Seifert fibred manifolds with orientable base orbifolds, we establish a necessary and sufficient condition for the Euler class of the normal bundle of the Seifert fibration to vanish. When the base orbifold is hyperbolic, we also provide a second proof of this condition from the perspective of discrete faithful representations of Fuchsian groups. As an application, we present infinitely many Seifert fibred rational homology spheres that admit co-oriented taut foliations but none with vanishing Euler class. In the context of the $L$-space conjecture, these provide examples of rational homology spheres that admit co-oriented taut foliations (and hence are not $L$-spaces) and have left-orderable fundamental groups yet none of the left-orders arise directly from the universal circle actions associated to co-oriented taut foliations.