The propagation of elastic waves is a complex phenomenon at stake in large array of applications. An interesting feature is that elastic materials supports both pressure and share waves, which can interact in an intricate manner. These two kinds of waves have different wave speed that are linked with the Lam\'e parameters $\lambda$ and $\mu$ describing the elastic properties of the propagation medium. In this work, we are interested in nearly incompressible materials (e.g. rubber) where the ratio $\lambda/\mu$ (or $\max \lambda/\min \mu$ for heterogeneous media) can be arbitrarily large. It physically means that the pressure waves travel much faster than the shear waves. We focus on the time-harmonic regime, and our goal is to establish stability estimates. Specifically, our objective is to bound the norm of the solution by the norm of the right-hand side explicitly in the frequency parameter, and in the ratio $\lambda/\mu$. In addition to their intrinsic interest, these stability estimates are also crucial to understand the accuracy and performance of numerical discretization of high-frequency problems, especially for elastic materials with a large difference of pressure and shear wave speeds. For a large class of problems (including scattering by a star-shaped obstacle) we are able to show that the stability constant relating the norm of the solution to the one of the right-hand side increases linearly with the frequency. This result is expected, since it already holds for the simpler case of scalar Helmholtz problems. We additionally show that the stability constant remains uniformly bounded as the ratio $\lambda/\mu$ tends to infinity. The estimates we derive are fully-explicit, and we therefore expect they can be useful later on in a priori and a posteriori error analysis of numerical discretizations.