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A flag on a manifold $M$ is an increasing sequence of foliations $\left({ℱ}_1,\cdots ,{ℱ}_p\right)$ on this manifold, where for each $i$, $dim{ℱ}_i=i$. The aim of this paper is to etablish that any flag of riemannian foliations $𝒟=$ $\left({ℱ}_1,\cdots ,{ℱ}_p\right)$ on a compact and connected manifold, lifts on the bundle of transverse direct orthonormal frames of ${ℱ}_p$ to a flag of transversally parallelizable foliations. This result permits us to obtain a classification of riemannian flags of a $n+1$-dimensional compact manifold for which the dimension of the structural...