There are a few exercises in Hartshorne about checking singularity of an affine curve. For example, $Y$ defined by $x^2 = x^4 + y^4$ over a field $k$ (with ${\mathrm{char}}k \neq 2$). This is easy.
However, I was wondering how easy it is to prove that it is a curve; or equivalently $f(x,y) = x^4 + y^4 - x^2$ is irreducible.
I just figured it out (Sorry to bother). If it is reducible then we have two choices:
(1) $y^4 + x^4 - x^2 = (y + f(x))(y^3 + g_2(x)y^2 + g_1(x)y + g_0(x))$. In this case either $f(x) = 0$ or $g_0(x) = 0$. Neither is possible as the polynomial is not in the ideal $\langle y \rangle$.
(2) $y^4 + x^4 - x^2 = (y^2 + f_1(x)y + f_0(x))(y^2 + g_1(x)y + g_0(x))$. Here comparing the terms either $f_1(x) = 0$ or $f_0(x) = 0$. In the first case $g_1(x) = 0 = g_0(x)$. In the second case $f_1(x) = 0$. None of them are possible as the polynomial is not in the ideal $\langle y^2 \rangle$.