I have already seen multiple times (e.g. in Neukirch ANT, Chapter 1.13) that if we have a one-dimensional noetherian ring $R$, then the spectrum Spec$(R)$ defines a smooth (i.e. non-singular) curve if and only if it is dedekind, i.e. integrally closed.
My question is now how to even start to prove this or in other words: What happens at singular points of a curve such that the ring is not integrally closed any more?
P.S. I know that localisation of Dedekind rings are discrete valuation rings (and thus regular local rings of dimension $1$). Can this be somehow used (i.e. is there a theorem stating that if we have vanishing partial derivatives, i.e. a singular point, then the corresponding local ring is not regular of dimension 1)?