  The study of bifurcations of differential-algebraic equations (DAEs) is the
topic of interest for many applied sciences, such as electrical engineering,
robotics, etc. While some of them were investigated already, the full
classification of such bifurcations has not been done yet. In this paper, we
consider bifurcations of quasilinear DAEs with a singularity and provide a full
list of all codimension-one bifurcations in lower-dimensional cases. Among
others, it includes singularity-induced bifurcations (SIBs), which occur when
an equilibrium branch intersects a singular manifold causing certain
eigenvalues of the linearized problem to diverge to infinity. For these and
other bifurcations, we construct the normal forms, establish the non-degeneracy
conditions and give a qualitative description of the dynamics. Also, we study
singular homoclinic and heteroclinic bifurcations, which were not considered
before.
