Since the Poincare conjecture is known in all dimensions any homotopy $n$-sphere is homeomorphic to $S^n$ and hence admits a smooth structure.

Any manifold of dimension $\le 3$ admits a smooth structure.

Kirby-Siebenmann famously showed that a closed manifold $M$ of dimension $>4$ admits a PL structure if and only if certain element of $H^4(M;\mathbb Z_2)$ vanishes. The element is called the Kirby-Siebenmann class. Thus any homology sphere of dimension $\neq 4$ admits a PL structure.

Kervaire noted on p.71 of Smooth homology spheres and their fundamental groups that any PL homology sphere of dimension $\neq 3$ admist a smooth structure (because it bounds a contractible PL manifold and all such manifolds are smoothable).

Thus the only remaining case is homology $4$-spheres. Some of them have large fundamental groups for which $4$-dimensional surgery is not yet known to work. It seems smoothability of such spheres is open. Note that any closed PL $4$-manifold admits a smooth structure.

Since the Poincaré conjecture is known in all dimensions any homotopy $n$-sphere is homeomorphic to $S^n$ and hence admits a smooth structure.

Any manifold of dimension $\le 3$ admits a smooth structure.

Kirby-Siebenmann famously showed that a closed manifold $M$ of dimension $>4$ admits a PL structure if and only if a certain element of $H^4(M;\mathbb Z_2)$ vanishes. The element is called the Kirby-Siebenmann class. Thus any homology sphere of dimension $\neq 4$ admits a PL structure.

Kervaire noted on p.71 of Smooth homology spheres and their fundamental groups that any PL homology sphere of dimension $\neq 3$ admits a smooth structure (because it bounds a contractible PL manifold and all such manifolds are smoothable).

Thus the only remaining case is homology $4$-spheres. Some of them have large fundamental groups for which $4$-dimensional surgery is not yet known to work. It seems smoothability of such spheres is open. Note that any closed PL $4$-manifold admits a smooth structure.