We propose a denoising method that has the property of preserving local regularity, in the sense of local H ̈lder exponent. This approach is fitted to the processing of irregular signals, and gives specially relevant results for those displaying a local form of scale invariance known as localisability. A wavelet decomposition is used to measure and control the local H ̈lder exponent. The main ingredient of the algorithm is an estimator (which is of independent interest) of the time-dependent cut-off scale beyond which wavelet coefficients are mainly due to noise. Based on local regularity estimated from information below the cut-off scale, these small-scale coefficients -which govern the texture- are cor- rected so that the H ̈lder exponent of the denoised signal matches the one of the original signal. The processing is only slightly more com- plex than classical wavelet coefficients thresholding, resulting in fast computing times. Numerical experiments show the good performance of this scheme on various localisable signals.