A motion of a mechanism is a curve in its configuration space (c-space). Singularities of the c-space are kinematic singularities of the mechanism. Any mobility analysis of a particular mechanism amounts to investigating the c-space geometry at a given configuration. A higher-order analysis is necessary to determine the finite mobility. To this end, past research leads to approaches using higher-order time derivatives of loop closure constraints assuming (implicitly) that all possible motions are smooth. This continuity assumption limits the generality of these methods. In this paper, an approach to the higher-order local mobility analysis of lower pair multiloop linkages is presented. This is based on a higher-order Taylor series expansion of the geometric constraint mapping, for which a recursive algebraic expression in terms of joint screws is presented. An exhaustive local analysis includes analysis of the set of constraint singularities (configurations where the constraint Jacobian has certain corank). A local approximation of the set of configurations with certain rank is presented, along with an explicit expression for the differentials of Jacobian minors in terms of instantaneous joint screws. The c-space and the set of points of certain corank are therewith locally approximated by an algebraic variety determined algebraically from the mechanism's screw system. The results are shown for a simple planar 4-bar linkage, which exhibits a bifurcation singularity and for a planar three-loop linkage exhibiting a cusp in c-space. The latter cannot be treated by the higher-order local analysis methods proposed in the literature.