Let $R$ be a complete DVR of mixed characteristic with fraction field $K$ of characteristic $0$ and residue field $k$ of characteristic $p>0$. Suppose that $\mathcal{X}$ is a normal $R$-curve such that $\mathcal{X}_{K}$ is smooth and integral but $\mathcal{X}_{k}$ is non-reduced and irreducible. I would like to characterise the (geometric) singular points of $\mathcal{X}$ namely, the closed points $x\in \mathcal{X}$ such that $\mathcal{O}_{\mathcal{X},x}$ is not regular.

In particular, it would be interesting to know:

1. When are the singular points of $\mathcal{X}$ a finite set?
2. Can one describe the singular points of $\mathcal{X}$ in terms of $k$-points that lift to $R$-points? i.e. it would seem for intersection theoretic reasons that an $R$-point $\sigma \in \mathcal{X}(R)$ should reduce to a $k$-point $\tilde{\sigma}$ that is singular for $\mathcal{X}$. However, should one expect that all singular points of $\mathcal{X}$ lie on the special fibre, and arise this way?

As a motivating example one could consider smooth affine curves $\mathcal{W}\subseteq \mathbb{A}^{2}_{\mathbb{Z}_{p}}$ given by an irreducible polynomial $f\in \mathbb{Z}_{p}[x,y]$ and the scheme-theoretic preimage under: $F:\mathbb{A}^{2}_{\mathbb{Z}_{p}}\rightarrow \mathbb{A}^{2}_{\mathbb{Z}_{p}}$ which maps $(x,y)\mapsto (x^{p},y^{p})$. Then the preimage $\mathcal{X}=F^{-1}(\mathcal{W})=\{f(x^{p},y^{p})=0\}$ has non-reduced special fibre: $\mathcal{X}_{\mathbb{F}_{p}}=\{\tilde{f}(x,y)^{p}=0\}$. Is there any way one could hope to determine if the (geometric) singular points of $\mathcal{X}$ are finite in number?