Here is a list of 3-manifolds $Y$ that are boundaries of exotic 4-manifolds https://arxiv.org/pdf/1901.07964.pdf
If either $Y$ or $-Y$ (i.e with reverse orientation) has a contact structure with non-trivial contact invariant.
If $Y$ or $-Y$ has weak symplectic filling.
If $Y$ bounds both positive and negative definite 4 manifolds.
In case of 1) and 2) those manifolds bounds simply-connected manifolds with infinitely many exotic structures and in case of 3rd case we cannot gurantee the simply-connected condition.
The above classes cover all Seifert fibered 3-manifolds, all 3 manifolds that admits Taut folitaion, all irreducible 3 manifolds with 1st Betti number strictly bigger than zero or $M\# -M$.
Conjecturally we covered all the irreducible 3-manifolds in the above 3 cases. One obstruction when dealing with reducible manifold is that most of the 4-manifolds invariant vanishes under connected sum. So it is still an open problem if all 3-manifolds bounds exotic 4-manifolds. Hope in some near future we (or someone else) will find some clever way to deal with all 3-manifolds.
In Theorem 1.13 above we gave a general construction which holds for every 3-manifolds (because all 3-manifolds admit contact structures). But we do not know how to prove that they all are not diffeomorphic in general.