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We propose a new multifractal formalism, based on histograms of wavelet coefficients. This new formalism yields a spectrum of H\"older and oscillating singularity exponents, and is not limited to concave spectra. We show that the corresponding upper bound holds with a minimal global regularity hypothesis. This upper bound is achieved for almost all realizations of a stochastic process called random wavelet series, obtained by drawing independently the wavelet coefficients with a general scale-dependent law.