Broadband frequency-selective fading channels usually have the inherent sparse nature. By exploiting the sparsity, adaptive sparse channel estimation (ASCE) algorithms, e.g., least mean square with reweighted L1-norm constraint (LMS-RL1) algorithm, could bring a considerable performance gain under assumption of additive white Gaussian noise (AWGN). In practical scenario of wireless systems, however, channel estimation performance is often deteriorated by unexpected non-Gaussian mixture noises which include AWGN and impulsive noises. To design stable communication systems, sign LMS-RL1 (SLMS-RL1) algorithm is proposed to remove the impulsive noise and to exploit channel sparsity simultaneously. It is well known that regularization parameter (REPA) selection of SLMS-RL1 is a very challenging issue. In the worst case, inappropriate REPA may even result in unexpected instable convergence of SLMS-RL1 algorithm. In this paper, Monte Carlo based selection method is proposed to select suitable REPA so that SLMS-RL1 can achieve two goals: stable convergence as well as usage sparsity information. Simulation results are provided to corroborate our studies.