The $d$ invariant coming from the rational grading can be seen as an enhancement of the torsion linking form on a rational homology $3$-sphere. Just like the regular torsion linking form of 3-manifolds, this gives an obstruction to embedding rational homology $3$-spheres in the $4$-sphere (or any homology $4$-sphere). Similarly, this gives obstruction to $spin^c$-cobordism. Sometimes these obstructions are completely new, so it's a genuinely novel application where there were no comparable tools beforehand.