This paper deals with speech enhancement methods using the wavelet transform. A cycle-spinning scheme and undecimated wavelet transform are used for denoising of speech signals, and then their results are compared with that of the conventional wavelet transform. We apply soft-thresholding technique for removing additive background noise from noisy speech. The symlets 8-tap wavelet and pyramid algorithm are used for the wavelet transform. Performance assessments based on average SNR, cepstral distance and informal subjective listening test are carried out. Experimental results demonstrate that both cycle-spinning denoising(CSD) method and undecimated wavelet denoising(CWD) method outperform conventional wavelet denoising(UWD) method in objective performance measure as welt as subjective listening test. The two methods also show less "clicks" that usually appears in the neighborhood of signal discontinuities.