Reference Blajer et al. in the nonholonomic constraints chapter where I discuss possible singularities

[moorepants]

Blajer, W., Schiehlen, W., & Schirm, W. (1994). A projective criterion to the coordinate partitioning method for multibody dynamics. Archive of Applied Mechanics, 64(2), 86–98. https://doi.org/10.1007/BF00789100