First off, many of your questions above about when to decide what curve to use, where to stop, where to mark data as outliers are all dependent on WHERE this data is coming from. For instance, a generalized rule such as the diag(S) might happen to apply to your case, but it sure isn't correct for many cases I have seen.If you give detailed information on the origins of the data, I might be able to offer more specific suggestions.
Being completely blind as to why you are doing this, I would suggest you start with fitting all available polynomial surfaces. and compare performance of each fit. Use an overfit measure like Cross Validation to decide when to stop increasing the degree of polynomial.
For example, lets say in your application area, 5% average error for the fit (notice this is not R^2) is acceptable. Since you are familiar with SVD might as well stick to principal components since that is generally better conditioned for regression. Try polynomial fits up to power 5th on the first and second principal components alone. If you get to 5% error or below like that, there you go you don't need the 3rd dimension. If you can't, try the same thing up to 5th power with all 3 components.
The above is just one particular scheme I made up to give you an idea, you can change a lot of decisions I made to fit your own problem.
I show something similar to this in the example for my regression code MultiPolyRegress, even if want to stick to the built in functions, the example might help.
