Let $k$ be a field. Let $X$ be a reduced $k$-scheme of finite type. If $X$ is geometrically reduced, then it is a basic result that $X$ has a regular point (i.e. the local ring at that point is regular). If $k$ is not perfect, then $X$ need not be geometrically reduced. I am looking for an example of a reduced variety over a non-perfect field $k$ which has no regular points at all (if such a thing exists?)
1$\begingroup$ Amusingly, there are examples of 1-dimensional (non-excellent) noetherian domains that are non-regular at all maximal ideals, examples given by Mel Hochster. Try Google with "Hochster" and "non-regular", or something like that (or maybe someone else will remember where this was previously mentioned on MO). $\endgroup$– BCnrdOct 1, 2010 at 16:35
I can just add a reference to Angelo's comment. If $X$ is a scheme locally of finite type over a field $k$ (or more generally over a complete local noetherian ring $k$), then $Reg(X)$ is open. Cf. EGA IV.6.12.5 and IV.6.12.8. The results seem to go back to Nagata and Zariski.
If $X$ is of finite type over a field $k$ and reduced, then Angelo's argument shows that $Reg(X)$ is non-empty as well. Hence $Reg(X)$ will also contain closed points in that case.
No, this is impossible. The regular locus of $X$ is open, and contains the generic point of each of its irreducible components, so it is dense in $X$.