Let $n$ be a positive integer and let $A$ be the subring of ${\mathbb C}[x,y]$ generated by $x,xy,...,xy^n$. Let $S=Spec(A)$. This is an affine surface, which is clearly singular if $n\neq 1$. Is this some sort of familiar surface? For example, is it normal? Does it have rational singularities? Can one construct an explicit resolution of singularities for it?

Let $\mathbb{F}_n$ be the Hirzebruch surface of index $n$. You have two open subsets on it isomorphic to $\mathbb{A}^1\times \mathbb{P}^1$, with a glueing map given by $$(t,[u:v])\mapsto (1/t,[ut^n:v])$$ The section $u=0$ has selfintersection $n$ and the section $v=0$ has selfintersection $n$. We remove the section $v=0$ and obtain an affine surface if $n>0$ (because the section is ample). The morphism to your surface given by $$(t,[u:v])\mapsto (\frac{u}{v},\frac{u}{v}t,\dots,\frac{u}{v}t^n)$$ on the first chart corresponds to $$(t,[u:v])\mapsto (\frac{u}{v}t^n,\frac{u}{v}t^{n1},\dots,\frac{u}{v}t,\frac{u}{v})$$ on the other chart. Hence the resolution of your singularity is just the complement in $\mathbb{F}_n$ of one section of selfintersection $n$, and the exceptional divisor is the section of selfintersection $n$. The surface obtained is called a Gizatullin surface, because of the very nice theorem of Gizatullin that asserts that the isomorphism class of the complement of an ample section in a Hirzebruch surface only depends on the square of the section (and not of the section or the Hirzebruch surface). 


Here is an alternative, perhaps more familiar way to recognize this surface. First consider the $\mathbb C$algebra embedding $$ \mathbb C[x,y]\hookrightarrow \mathbb C[z,w,z^{1}] $$ given by $$ x\mapsto z^n \qquad y\mapsto wz^{1}. $$ Since $z,w$ are algebraically independent, this is clearly an embedding. It is easy to see that via this embedding the image of the subring $\mathbb C[x,xy,\dots,xy^n]\subset \mathbb C[x,y]$ maps isomorphically to the subring $\mathbb C[z^n,z^{n1}w,\dots,zw^{n1},w^n]$. Spec of this latter ring and hence the surface $S$ in the question is the (affine) cone over the (projective) rational normal curve of degree $n$ in $\mathbb P^n$. From this description it is easy to answer the additional questions and even more about the singularity:
... and the same argument works over any algebraically closed field, not just $\mathbb C$. 

