  We give the complete solution to the local diffeomorphism classification
problem of generic singularities which appear in tangent surfaces, in as wider
situations as possible. We interpret tangent geodesics as tangent lines
whenever a (semi-)Riemannian metric, or, more generally, an affine connection
is given in an ambient space of arbitrary dimension. Then, given an immersed
curve, or, more generally a directed curve or a frontal curve which has
well-defined tangent directions along the curve, we define the tangent surface
as the ruled surface by tangent geodesics to the curve. We apply the
characterization of frontal singularities found by Kokubu, Rossman, Saji,
Umehara, Yamada, and Fujimori, Saji, Umehara, Yamada, and found by the first
author related to the procedure of openings of singularities.
