It is known that every oriented plane field on a closed 3-manifold is homotopic to an integrable one. However, this no longer holds if one requires the foliation to be taut. This leads naturally to the question of which second cohomology classes can arise as the Euler classes of co-oriented taut foliations on a given 3-manifold M. When M is a rational homology sphere, the second cohomology group is finite, and the zero class plays a distinguished role.

In this talk, we present infinitely many rational homology 3-spheres, including small Seifert fibred, hyperbolic, and toroidal examples, that admit co-oriented taut foliations but do not admit any with vanishing Euler class. We will also discuss the implications of these examples in the context of the L-space conjecture. This is joint work with Steve Boyer, Cameron Gordon and Duncan McCoy.