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



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> Are there any non-compact, non-smoothable manifolds?
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Are there non-compact, non-smoothable manifolds?
The Cairns-Hirsch 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 non-smoothable for $n \geq 1$.


