In the situation when smooth/k coincides with regularity (for a finite-type k scheme edit: k also perfect, thanks Remy), I think this should be true (?). But I am not sure about the situation for a general scheme.
Maybe there is a scheme consisting only of singular (non-regular) points? Oh, there is, I think: $k[x]/x^2$. But this doesn't answer my question about openness.
By regularity I mean that local ring is a regular local ring, i.e. the dimension of the Zariski tangent space is the same as the dimension of the local ring.
No. According to this mathoverflow post, there is an example of an affine Noetherian integral scheme of dimension 1 whose regular locus is not open, see Exposé XIX of the volume "Travaux de Gabber" in Astérisque 363-364.
See comments by Remy and Rieux for nice sufficient conditions for the regular locus to be open.