  This study examines the formulation of a singularity theorem for timelike
curves including torsion, and establishes the foundational framework necessary
for its derivation. We begin by deriving the relative acceleration for an
arbitrary congruence of timelike curves. The resulting ``deviation equation''
offers an alternative pathway to the well-known Raychaudhuri equation with
torsion. Conjugate points are then introduced and analyzed in relation to the
behavior of the scalar expansion. Together with the sensible requirement of
hypersurface orthogonality, the Raychaudhuri equation is examined for several
specific cases of torsion that are prominent in the literature. Our findings
indicate that a totally antisymmetric torsion tensor does not influence the
behavior of the congruence of timelike curves. Finally, we formulate a
singularity theorem for timelike curves and highlight the critical requirement
of non-autoparallel curves.
