As suggested by Phil on the mailing list, I'm providing a patch to the javadoc of DecompositionSolver.getInverse() to clarify what it does. It also correctly moves the "@throws SingularMatrixException" to implementations that do throw this exception; not all do.
Tests already cover most of the behavior asserted by the documentation. I added one to show that the SVD does not reject singular matrices. There should be a test for EigenDecomposition here too, but in the course of adding it I found another apparent small bug in its behavior with singular matrices. So I will provide that separately (it actually involves no doc changes, and docs are the topic here.)