Is there a classification of singularities from $S^2 \to \mathbb{R}^2$ ? The critical points of the map $(x,y) \mapsto (f_1(x,y),f_2(x,y))$ where the matrix:

\[ \left[\begin{array}{cc}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y}\\\\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y} \end{array} \right] \]

has less than full rank. Locally, can we draw a picture of what singularities look like?

In the case of maps $S^2 \to \mathbb{R}^1$, we just get the critical points where $f(x,y) = f(x_0,y_0) + (x-x_0,y-y_0)^T (D^2 f )(x-x_0,y-y_0)$