I've heard once assertions of the sort:

1. Let there is a Riemann metric (not very smooth, say of a class $C^1$ or $C^2$ or may be $C$?) in a neibourhood of a point on a manifold. Then it is possible to choose such coordinates what the metric is $C^\infty$ or even analitic in them.
2. In case of 3-dimentional manifolds it is possible to choose such coordinates globally, so the manuifold becomes a smooth one. In case of higher dimentions $n\ge4$ it is not true.

Are those assertions true? I've heard them some time ago and not sure I remember all the details. Is it a well-known thing? Are there some detailed references?