We provide a comprehensive study of uniformly left bounded (resp. left-right bounded) orthonormal bases in GNS spaces of infinite-dimensional von Neumann algebras in the framework of both faithful normal states and f.n.s. weights. There are two issues to consider: one concerning the existence of such bases and the other concerning the bound in operator norm of the left (resp. left and right) multiplication operators associated to such bases. We provide necessary and sufficient conditions on a closed subspace of a GNS space to guarantee the existence of an orthonormal basis of uniformly left bounded (resp. left-right bounded) vectors. In the context of states, while a basis of the first kind exists for all GNS spaces, B(ℓ2) is excluded for a basis of the latter kind. However, in the context of weights, there are no such obstructions. In the context of weights, the GNS space of every infinite-dimensional von Neumann algebra admits a uniformly left and right bounded orthonormal basis such that the aforesaid bound is arbitrarily small.