In what follows assume that the base field is $\mathbb{C}$

Background: Let $C$ be an irreducible plane algebraic curve, $S$ the set of singular points. There exists a Riemann surface $\tilde{C}$ and a holomorphic mapping $\sigma : \tilde{C} \to C$ such that $\sigma^{-1}(S)$ is finite and $\sigma : \tilde{C} \backslash \sigma^{-1}(S) \to C \backslash S$ is a biholomorphism.

We say $(\tilde{C}, \sigma)$ is the normalisation of $C$.

Question: I know the normalisations of some smooth plane algebraic curves (e.g the normalisation of a non singular elliptic curve is a torus)

Can anyone give me some examples of the normalisations of singular plane algebraic curves? What I am ideally looking for is an example of a singular plane algebraic curve and the corresponding Riemann surface and map.