You're welcome. I am glad I could be of help.

I think this is also Thm 23.7(i) in Matsumura. The statement there is stated for a local flat homomorphism, but regularity and flatness are local properties.

@abx Thanks for the answer. Sorry, but I don't get your hint, can you spell out a bit more? (Meanwhile I figured out another proof by showing that $J$ satisfies the universal property of $\mathrm{Alb}(C^{(n)})$ but I still would like to understand your approach). I tried to use that $\mathrm{Alb}(X)=\mathrm{Pic}^0(X)^{\vee}$ and the exponential sequence, but I don't see where the isomorphism on $H_1(-,\mathbb{Z})$ comes from and how to apply it.

@abx Sorry for the intrusion, but why is $\mathrm{Alb}(C^{(n)}\times C^{(n)})\cong J\times J$? Is in general, $\mathrm{Alb}(C^{(n)})\cong J$? Where does it come from?

You're welcome. I think it really depends whom do you ask to. Being more a commutative algebraist guy I'd like to think "singular" = "not regular", but I guess other people would prefer "not smooth".

Dear Takumi, thank you for your nice and precise answer. It looks correct to me. The key point here is really the fact that finitely generated algebras over a field are $G$-rings. I have not thought about it. Concerning your question about the "standard" definition of isolated singularity. Personally, when I say isolated singularity I mean the first one. Or more algebraically, I say that a local ring $(R,\mathfrak{m})$ has an isolated singularity if $R_{\mathfrak{p}}$ is regular for any prime ideal $\mathfrak{p}\neq\mathfrak{m}$. I am not sure whether this is "standard" or not.

Yes, that paper is also related. However, there we focus more on some geometric consequences of knowing explicit equations for the parameter space of points supported on a r.n.c. I think that the first one is more concerned with Speyer's question. Anyway thank you for pointing it out, and sorry for my late answer!