In this paper, we begin to consider viscous and wettability effects in the interfacial stability of fluid dynamic bearings in hard-disk drives. As a canonical problem, the linear stability of a sheared meniscus between two parallel plates is investigated when inertial effects are neglected. Through an asymptotic analysis, we prescribe the appropriate boundary conditions on the interfacial contact line which regularizes the contact-line singularity due to the application of no-slip. We find that for sufficiently large capillary numbers, two neutrally stable modes are always found, and the next least stable modes correspond to temporally-decaying standing waves, whose average speed is the same as the average base-state flow velocity. Although all modes are stable, we discuss the impact of these results when inertial effects are pertinent.