For a noetherian ring R, Serre's criterion for normality states that R is normal if and only if R satisfies conditions R1 and S2, where R1 is regularity in codimension one, and S2 is Serre's condition that every prime P of codimension at least 2 satisfies depth R_P \geq 2. Likewise, there's a similar condition for whether or not R is reduced: R is reduced iff R satisfies R0 and S1. Following the pattern, it seems like Rn and S(n+1) should be equivalent to some desirable property for rings.
Now, there's a nice and canonical way to take any ring and create a reduced (or normal) ring out of it, and this is something we should not expect to extend to higher dimensions, since we know that resolving singularities is (a) hard and (b) not very canonical. That aside, surely we ought to be able to say something nontrivial about the next case, ie, R2 and S3?