The proximity relations inherent in triangulations of geometric data can be exploited in the implementation of nearest-neighbour search procedures. This is relevant to applications such as terrain analysis, cartography and robotics, in which triangulations may be used to model the spatial data. Here we describe neighbourhood search procedures within constrained Delaunay triangulations of the vertices of linear objects, for the queries of nearest object to an object and the nearest object to an arbitrary point. The procedures search locally from object edges, or from a query point, to build triangulated regions that extend from the source edge or point by a distance at least equal to that to its nearest neighbouring feature. Several geographical datasets have been used to evaluate the procedures experimentally. Average numbers of edge–edge distance calculations to find the nearest line feature edge disjoint to another line feature edge ranged between 15 and 39 for the different datasets examined, while the average numbers of point–edge distance calculations to determine the nearest edge to an arbitrary point ranged between 7 and 35.