Can't you compute the singular locus using the Jacobian criterion?

Yes it is this product. The reason is also in the Eisenbud Sturmfels paper. In their terminology you are asking: How many saturations does the character defined by your Laurent lattice ideal have. The answer is: The order of the finite group $\text{sat}(L)/L$ and the Smith normal form computes a canonical embedding of L into $ZZ^d$ from which you extract it.

I'm writing a symbolic computation answer but you may want to try numerical algebraic geometry software such as Bertini.

This is counting lattice points in polytopes. You can google for this beautiful theory. Concrete instances such as the one you are giving can be solved with the software Latte: math.ucdavis.edu/~latte

It follows from the before-mentioned Mayhew-Newman-Whittle theorem which is more precise: For any real representable matroid M there exists a forbidden minor for real representability that has M as a minor.

@Hailong + Patricia: It is the ideal generated by the union, so prime avoidance can't be applied necessarily. As in the example: $(x+y) \subset (x) + (y)$.

Yes that would be possible using ideals generated by variables, I think. It's probably unrelated, though. For simplicity we may assume that the original ideals are all pairwise containment incomparable. (that is not the case in the non-matroid example, but one could come up with another example). Then what is the poset you are thinking of? Is it the poset of all possible sums of those ideals?