<p>The mortar method is a powerful technique to enforce constraints between non-conforming discretizations by introducing a set of Lagrange multipliers on the connecting interface. Usually, the multipliers are not obtained explicitly because they can be eliminated with the aid of the so-called mortar interpolation operator. However, their explicit computation becomes essential when the contact constraint is governed by some non-linear law, and in this situation it is necessary to guarantee that discrete spaces of the primary variables and multipliers are inf-sup stable. In this work, we investigate the issue of inf-sup stability when using various families of piecewise linear and piecewise constant multipliers. The focus is on the role of the mesh resolution and the enforcement of boundary conditions, which are important factors in practical applications. Then, we develop a stabilized formulation for piecewise-constant multipliers inspired by the framework of minimal stabilization. The effectiveness of the proposed approach is demonstrated through numerical benchmarks and examples.</p>