Broadband signal transmission over frequency-selective fading channel often requires accurate channel state information at receiver. One of the most attracting adaptive channel estimation (ACE) methods is least mean square (LMS) algorithm. However, its performance is often degraded by random scaling of input training signal. To overcome this degradation, in this paper we consider the use of standard least mean square/fourth (LMS/F) algorithm. Since the broadband channel is often described by sparse channel model, such sparsity could be exploited as prior information. First, we propose an adaptive sparse channel estimation (ASCE) method with zero-attracting LMS/F (ZA-LMS/F) algorithm by introducing an ℓ1-norm sparse constraint into the cost function. Then, to exploit the sparsity more effectively, an improved ASCE with reweighted zero-attracting LMS/F (RZA-LMS/F) algorithm is proposed. For different channel sparsity, we propose a Monte Carlo method for a regularization parameter selection in RA-LMS/F and RZA-LMS/F to achieve better steady-state estimation performance. Simulation results show that the proposed ASCE methods achieve better estimation performance than the conventional one.