A combined numerical and analytical study is presented of the linear stability problem for resistive tearing modes in the presence of perpendicular and parallel equilibrium flows. Detailed information is given about growth rates and the shape of the marginal stability curve. In the case of purely perpendicular flows, a region in parameter space is found where the growth rate is complex, and the region of instability is larger than previously recognized. An exact analytic solution is found for parallel flows when the plasma response is dominated by viscous forces.