We apply optical trapping techniques in a magnetorheological (MR) suspension, allowing us to directly measure the mechanical properties of single dipolar chains, such as the rupturing tensions and strains under tensile and lateral deformations. Our results are in excellent agreement with calculations of the rupturing tensions using a self-consistent point dipole model of the particle interaction that accounts for induction and multibody effects along the chain. Additionally, we observe the annealing of chain defects under an applied stress, such as the inclusion of neighboring particles into the chain. The micromechanical properties of single chains offers important insight into magnetorheology and electrorheology, especially the yield stress behavior.