There do exist manifolds which do not admit any smooth structure at all. But the only examples I've heard of are all compact.
Are there any noncompact, nonsmoothable manifolds?
There do exist manifolds which do not admit any smooth structure at all. But the only examples I've heard of are all compact.



The CairnsHirsch theorem says that a PL manifold $M$ is smoothable if and only if $M\times \mathbb{R}$ is smoothable, so you can take $M$ to be any one of the known compact, PL examples such as Kervaire's manifold and then $M\times\mathbb{R}^n$ is nonsmoothable for $n \geq 1$. 

