A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, should one demand $S$ to be excellent? Will anything change if we want singular points not to be dense in any finite type reduced $S$-scheme? 


I would also be gratefull for any ('bad' or 'good') examples.



When singular points of a reduced scheme are not dense in it?
For examples with dense singular locus, see William J. Heinzer and Lawrence S. Levy: Domains of Dimension 1 with Infinitely Many Singular Maximal Ideals, Rocky Mountain J. Math. (2007), 203-214. Their examples are affine and noetherian.


