Let (R,m) be a Noetherian local ring, I, J two ideals of R, and M a finitely generated R−module. It is shown that the set Ass_R(H^{d−1}_I(N_n))∪{m} is stable for large n, where we use N_n to denote R−module

J^nM/J^{n+1}M or R−module M/J^nM and d is the eventual value of \dim(N_n). We also give a counter-example for stability of certain sets of associated prime ideals of \Ext−modules and of attached prime ideals of local cohomology modules w.r.t. \m.