Stability of delta-sigma loops is pretty complicated. I have run into various issues designing them but do not claim to be an expert, especially for audio converters that tend to be much more complex than the RF types I have usually designed (or helped design). Two popular introductory references are by John Candy & Gabor Temes (a collection of tutorials and IEEE papers) and the more recent book by Steven Norsworthy et. al. (see below). They are both pretty old now and there are a ton of other books, natch.
The oscillation will show up in spectral analysis but there are several types of oscillatory mechanisms. Some are signal-dependent, some relate to the DC level, and some are related to basic loop stability. So, you have to know what you are looking for, and they are not always immediately obvious.
Most delta-sigma loops are digital or sampled-analog (I have also designed continuous-time (CT) modulators and they add more worms to the can). Sampling is a non-linear phenomenon and so the phase shift is nonlinear and related to the architecture (including modulator/demodulator loop design) and sampling rate. Since a delta-sigma modulator is a feedback system they can get very complicated in design and analysis. Multiple bits, multiple loops, cascading loops, etc. Stability can be a nightmare, and simulations to ensure unconditionally stability of a nonlinear sampled system are not fun. And likely to miss that one magic combination that the real world finds immediately after power-up of the fabricated device...
That said, a lot of those issues have been addressed over the years, and I would not have expected them to be significant concerns, at least not stability, these days. But then I have expected a lot of things that haven't really worked out for me...
HTH - Don
Oversampling Delta-Sigma Data Converters: Theory, Design, and Simulation
James C. Candy (Editor), Gabor C. Temes (Editor)
Publisher: Wiley-IEEE Press; 1 edition (September 2, 1991)
Delta-Sigma Data Converters: Theory, Design, and Simulation
Steven R. Norsworthy (Editor), Richard Schreier (Editor), Gabor C. Temes (Editor)
Publisher: Wiley-IEEE Press; 1 edition (October 28, 1996)