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.