Time-domain models of dynamical systems are formulated in many applications in terms of differential-algebraic equations (DAEs). In the linear time-varying context, certain limitations of models of the form E(t)x'(t) + B(t)x(t) = q(t) have recently led to the properly stated formulation A(t)(D(t)x(t))' + B(t)x(t) = q(t), which allows for explicit descriptions of problem solutions in regular DAEs with arbitrary index, and provides precise functional input-output characterizations of the system. In this context, the present paper addresses critical points of linear DAEs with properly stated leading term; such critical points describe different types of singularities in the system. Critical points are classified according to a taxonomy which reflects the phenomenon from which the singularity stems; this taxonomy is proved independent of projectors and also invariant under linear time-varying coordinate changes and refactorizations. Under certain working assumptions, the analysis of such critical problems can be carried out through a scalarly implicit decoupling, yielding a singular inherent ODE. Certain harmless problems for which this decoupling can be rewritten in explicit form are characterized. Some electrical circuit applications, including a linear time-varying analogue of Chua's circuit, are discussed for illustrative purposes.