All Dedekind domains of characteristic 0 and all local Noetherian rings of dimension at most 1 are J-2 rings.
The family of J-2 rings is closed under taking localizations and finitely generated algebras.
For an example of a Noetherian domain that is not a J-0 ring, take R to be the subring of the polynomial ring k[x1,x2,...] in infinitely many generators generated by the squares and cubes of all generators, and form the ring S from R by adjoining inverses to all elements not in any of the ideals generated by some xn.
More precisely S has a cusp singularity at every closed point, so the set of non-singular points consists of just the ideal (0) and contains no nonempty open sets.
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