As materials with broken translational symmetry, defects in smectic liquid crystals do not follow the traditional homotopy theoretic classification scheme, and a more geometrical approach is required. Using methods from singularity theory we study the topological classification and combination rules for point and line defects in two and three dimensional smectics. We give a local classification of defect structures for both point and line defects using a graph theoretical representation, and construct the path-dependent rules for defect combinations in terms of certain binary operations on the graphs. In particular, our work shows that defect morphology is rich in three dimensional smectics, with traditional invariants failing to distinguish many topologically distinct defects.