In this work, we investigate the spectrum of singularities of random stable trees with parameter $γ\in(1,2)$. We consider for that purpose the scaling exponents derived from two natural measures on stable trees: the local time $\ell^a$ and the mass measure $\textbf{m}$, providing as well a purely geometrical interpretation of the latter exponent. We first characterise the uniform component of the multifractal spectrum which exists at every level $a>0$ of stable trees and corresponds to large masses with scaling index $h\in[\tfrac{1+γ}γ,\tfracγ{γ-1}]$ for the mass measure (or equivalently $h\in [\tfrac{1}γ,\tfrac{1}{γ-1}]$ for the local time). In addition, we investigate the distribution of vertices appearing at random levels with exceptionally large masses of index $h\in[0,\tfrac{1+γ}γ)$. Finally, we discuss more precisely the order of the largest mass existing on any subset $\mathcal{T}(F)$ of a stable tree, characterising the former with the packing dimension of the set $F$.