One of the major findings from multimodal neuroimaging studies in the past decade is that the human brain is anatomically and functionally organized into large-scale networks. In resting state fMRI (rs-fMRI), spatial patterns emerge when temporal correlations between various brain regions are tallied, evidencing networks of ongoing intercortical cooperation. However, the dynamic structure governing the brain's spontaneous activity is far less understood due to the short and noisy nature of the rs-fMRI signal. Here, we develop a wavelet-based regularity analysis based on noise estimation capabilities of the wavelet transform to measure recurrent temporal pattern stability within the rs-fMRI signal across multiple temporal scales. The method consists of performing a stationary wavelet transform to preserve signal structure, followed by construction of "lagged" subsequences to adjust for correlated features, and finally the calculation of sample entropy across wavelet scales based on an "objective" estimate of noise level at each scale. We found that the brain's default mode network (DMN) areas manifest a higher level of irregularity in rs-fMRI time series than rest of the brain. In 25 aged subjects with mild cognitive impairment and 25 matched healthy controls, wavelet-based regularity analysis showed improved sensitivity in detecting changes in the regularity of rs-fMRI signals between the two groups within the DMN and executive control networks, compared with standard multiscale entropy analysis. Wavelet-based regularity analysis based on noise estimation capabilities of the wavelet transform is a promising technique to characterize the dynamic structure of rs-fMRI as well as other biological signals.