Resistive REBCO Fault Current Limiter (FCL) is a promising solution to the fault current issue in electric grids, which is not fully satisfactorily solved. But the cost of the Superconducting (SC) REBCO conductor should be lowered to facilitate the deployment of this innovative device. What we refer to as REBCO conductor is usually a bare REBCO tape bonded to a stabilizer. The design of the stabilizer is fundamental to optimize the conductor and lower the SFCL cost. The goal is to limit the temperature rise for any fault conditions, notably for any amplitude of the prospective fault current. The specificities of SC tapes, including critical current inhomogeneities along the length, must be considered in the design. Here we study various stabilizer configurations. First the case of “dielectric stabilizer”, electrically insulating is presented. The theoretical performances are interesting in terms of electric field under limitation, but the low thermal diffusivity remains an issue, as well as the implementation. Some experimental investigations are shown. The results are compared to classical metallic stabilizer designs, which are simpler to implement. To enhance the electric field under limitation, a variant of metallic stabilizer is then introduced, using a corrugated structure. A design showing an electric field under limitation of 200 V/m (clearing time of 50 ms) is presented. Further optimizations and higher electric fields seem achievable even if practical implementation remains very challenging.