Classifcation of non-linear regressions based on their shapes
As a quick brainstorming idea, you could create new attributes based on your original ones. 
Let's say you have a 2D feature set x and y; so, you could test if adding x2, y2 , x•y, x3... as new attributes, and applying regularization, gives you as important features the ones you expect for each type of regression (test it with hand made examples for exponential regression, logarithmic, etc)