Channel estimation problem is one of the key technical issues for broadband multiple-input-multiple-output (MIMO) signal transmission. To estimate the MIMO channel, a standard least mean square (LMS) algorithm was often applied to adaptive channel estimation because of its low complexity and stability. The sparsity of the broadband MIMO channel can be exploited to further improve the estimation performance. This observation motivates us to consider adaptive sparse channel estimation (ASCE) methods using sparse LMS (ASCE-LMS) algorithms. However, conventional ASCE methods have two main drawbacks: (i) sensitivity to random scaling of training signal and (ii) poor estimation performance in low signal-to-noise ratio (SNR) regime. The former drawback is tackled by proposing novel ASCE-NLMS algorithms. ASCE-NLMS mitigates interference of random scale of training signal and therefore it improves its algorithm stability. It is well-known that stable sparse normalised least-mean fourth (NLMF) algorithms can achieve better estimation performance than sparse NLMS algorithms. Therefore the authors propose an improved ASCE method using sparse NLMF algorithms (ASCE-NLMF) to improve the estimation performance in low SNR regime. Simulation results show that the proposed ASCE methods are shown to achieve better performance than conventional methods, that is, ASCE-LMS by computer simulations. Also, the stability of the proposed methods is confirmed by theoretical analysis.