When redundant constraints are present in a rigid body mechanism, only selected (if any at all) joint reactions can be determined uniquely, whereas others cannot. Analytic criteria and numerical methods of finding joints with uniquely solvable reactions are available. In this paper, the problem of joint reactions solvability is examined from the point of view of selected numerical methods frequently used for handling redundant constraints in practical simulations. Three different approaches are investigated in the paper: elimination of redundant constraints; pseudoinverse-based calculations; and the augmented Lagrangian formulation. Each method is briefly summarized; the discussion is focused on techniques of handling redundant constraints and on joint reactions calculation. In the case of multibody systems with redundant constraints, the rigid body equations of motion are insufficient to calculate some or all joint reactions. Thus, purely mathematical operations are performed in order to find the reaction solution. In each investigated method, the redundant constraints are treated differently, which—in the case of joints with nonunique reactions—leads to different reaction solutions. As a consequence, reactions reflecting the redundancy handling method rather than physics of the system are calculated. A simple example of each method usage is presented, and calculated joint reactions are examined. The paper points out the origins of nonuniqueness of constraint reactions in each examined approach. Moreover, it is shown that one and the same method may lead to different reaction solutions, provided that input data are prepared differently. Finally, it is demonstrated that—in case of joints with solvable reactions—the obtained solutions are unique, regardless of the method used for redundant constraints handling.