The relation between the singularities of multifractal energy spectral measures and the behaviours of the Green function is studied in the framework of a tight-binding Hamiltonian. If the measure mu (E) has a seating behaviour at energy E of the form Delta mu (E)= mu (E+ delta )- mu (E- delta ) varies as delta ^{alpha (E)}, it is proved that the imaginary part of the Green function P(E, epsilon ) scales as P(E, epsilon ) varies as epsilon ^{beta (E)} with beta (E)= alpha (E)+1, the reverse being also true. This is exemplified in the case of the density of states and the local density of states of the one-dimensional Fibonacci quasicrystalline chain.