Vector fields on a sphere

Question

I am interested in what smooth vector fields on spheres look like. I am aware of the hairy ball theorem prohibiting the existence of nowhere vanishing smooth vector fields on a sphere, but I would like to permit zeros of the vector field. Are there any theorems on the number of zeros are allowed? Ultimately I would like to be able to write down the general form of a smooth vector field as a sum of basis functions (much like sines and cosines provide a basis for piecewise continuous functions on an interval). Any help e.g. answers with explanations or relevant references would be greatly appreciated. Thanks.