| Summary: | In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the <i>osculating developable surface</i> of the timelike surface along the curve. The ruling of the osculating developable surface is parallel to the <i>osculating Darboux vector field</i>. The main goal of this paper is to classify the singularities of the osculating developable surface. To this end, two new invariants of curves are defined to characterize these singularities. Meanwhile, we also research the singular properties of osculating developable surfaces near their lightlike points. Moreover, we give a relation between osculating Darboux vector fields and normal vector fields of timelike surfaces along curves from the viewpoint of Legendrian dualities. Finally, some examples with symmetrical structures are presented to illustrate the main results.
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