We present a robust method of estimating an effective H\"older exponent locally at an arbitrary resolution. The method is motivated by the multiplicative cascade paradigm, and implemented on the hierarchy of singularities revealed with the wavelet transform modulus maxima tree. In addition, we illustrate the possibility of the direct estimation of the scaling spectrum of the effective H\"older exponent, and we link it to the established partition functions based multifractal formalism. We motivate both the local and the global multifractal analysis by showing examples of computer generated and real life time series
