This paper analyzes the representation theoretic stability, in the sense of Thomas Church and Benson Farb, of the rank-selected homology of the Boolean lattice and the partition lattice, proving sharp uniform representation stability bounds in both cases. It proves a conjecture of the first author and Reiner by giving the sharp stability bound for general rank sets for the partition lattice. Along the way, a new homology basis sharing useful features with the polytabloid basis for Specht modules is introduced for the rank-selected homology and for the rank-selected Whitney homology of any geometric lattice, resolving an old open question of Björner. These bases give a matroid theoretic analogue of Specht modules.