The authors investigate how stable (in very natural settings) a sphere packing can be. A sphere packing is called finitely stable if for every integer \(n \geq 1\) each set of \(n\) balls is fixed by its neighbors, and uniformly stable if for a sufficiently small \(\varepsilon>0\) every finite rearrangement of the balls where no ball is moved more than \(\varepsilon\) is the identity rearrangement. It is easy to show that a uniformly stable packing is finitely stable. In Section 2 the authors develop a method for checking if certain packings are uniformly stable. Necessary and sufficient conditions for finite stability are explained in Section 4. Certain well known packings such as the lattice packings \(D_d\) and \(A_d\) in \(E^d\), \(d \geq 3\) are shown to be uniformly stable. On the other hand, it is constructively proved in Section 3 that the densest cubic lattice packing in \(E^d\), \(d \geq 2\), is not uniformly stable while it is finitely stable by Corollary 4.1.