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This paper investigates the fractal structure of China's stock market by calculating the multifractal singularity spectrum and comparing the scaling behavior of the bubble phase of eight abnormal volatilities with that of normal fluctuation on the timeline. We find robust evidence that the Shanghai Stock Exchange Composite Index has multifractal features in the bubble and normal fluctuation periods, where the higher multifractality is associated with a bubble and more unstable market. The short-sighted administrative policies cause over-supply of intervention, which enhances the multifractality and increases the instability of the stock market. The multifractal parameter set (α 0 , Δ α , − B) might be used as a quantifier to characterize the status of the stock market. A policy aimed to improve the stability of the stock market should be devoted to optimizing the parameter set. • A multifractal process for the SSECI both in bubble and normal status are supported. • The parameter set (α 0 , Δ α , − B) distinguish the bubble from normal status well. • The marketization degree has a significant impact on the spectra of the SSECI bubbles. [ABSTRACT FROM AUTHOR]