Singular points For an algebraic variety $Y\subset\mathbb{C}^d$ defined by
$$ Y=\{z\in\mathbb{C}^d : \ f_1(z)=\dots=f_r(z)=0\} $$
we say that $y\in Y$ is a singular point if the rank of the complex Jacobian of $f=(f_1,\dots,f_r)$ at $y$ is less than $r$.
For example let $Y=\{(z_1,z_2)\in\mathbb{C}^2 : \ f(z)=0\}$, where $f(z_1,z_2)=z_1^2$. 
Then the rank of $f$ is that of $(0,2z_1)$, so only the origin is a singular point.
Am I wrong? Otherwise I am puzzled: this seem to depend on the choice of charts...

In fact your definition is not the correct one. One has to take the Jacobian with respect to a generating system of the vanishing ideal (which is a radical ideal). In your example, the vanishing ideal is $(z_1)$ (and not $(z_1^2)$), and the Jacobian is $(1,0)$ in every point. The subvariety is just a copy of the affine line, hence nonsingular.
The correct definition of a singular point does not depend on charts, and it can be formulated by means of the local ring. You can find all the details in every text on algebraic geometry.
