The multifractal framework relates the scaling properties of turbulence to its local regularity properties through a statistical description as a collection of local singularities. The multifractal properties are moreover linked to the multiplicative cascade process that creates the peculiar properties of turbulence such as intermittency. A comprehensive estimation of the multifractal properties of turbulence from data analysis, using a tool valid for all kind of singularities (including oscillating singularities) and mathematically well-founded, is thus of first importance in order to extract a reliable information on the underlying physical processes. The wavelet leaders yield a new multifractal formalism which meets all these requests. This paper aims at describing it and at applying it to experimental turbulent velocity data. After a detailed discussion of the practical use of the wavelet leader based multifractal formalism, the following questions are carefully investigated: (1) What is the dependence of multifractal properties on the Reynolds number? (2) Are oscillating singularities present in turbulent velocity data? (3) Which multifractal model does correctly account for the observed multifractal properties? Results from several data set analysis are used to discuss the dependence of the computed multifractal properties on the Reynolds number but also to assess their common or universal component. An exact though partial answer (no oscillating singularities are detected) to the issue of the presence of oscillating singularities is provided for the first time. Eventually an accurate parameterization with cumulants exponents up to order 4 confirms that the log-normal model (with $c_2 = -0.025\pm 0.002$) correctly accounts for the universal multifractal properties of turbulent velocity.