The relevant question for eigenproblems is not whether the original matrix is ill-conditioned (nearly singular), but whether its eigenvector matrix is ill-conditioned (nearly singular, i.e. nearly parallel eigenvectors, i.e. the matrix is nearly defective). In this case the whole concept of an eigenvector is problematic — the eigenvectors tend not to be a very useful basis.
Can you give a concrete example of what you are trying to do? What do you need the eigenvectors for?