The paper is devoted to study the spectrum of bounded linear operators on Banach spaces. The main result given in Theorem 3.7 is that the spectrum of a power bounded linear operator on a superreflexive Banach space, situated on the unit circle, is countable if and only if this operator is superstable. The latter notion is introduced by using ultrapower technique, see \textit{J. Stern} [Trans. Amer. Math. Soc. 240, 231-252 (1978; Zbl 0402.03025), \textit{S. Heinrich}, J. Reine Angew. Math. 313, 72- 104 (1980; Zbl 0412.46017) and \textit{C. W. Henson} and \textit{L. C. Moore}, Lect. Notes Math. 283, 27-112 (1983; Zbl 0511.46070)]. To obtain their main result, the authors have proved new interesting assertions in the theory of ultrapowers, in the geometry of Banach spaces and in the theory of operators on Banach lattices.