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We study the properties of bright polariton solitons existing above the excitonic resonance in semiconductor microcavities operating in the strong-coupling regime. Saturation of the exciton-photon coupling due to electron-hole pair screening plays the crucial role in their formation and essentially determines the properties of these localized structures. Taking into account the finite mass of coherent excitons leads to modulational instabilities due to the generation of free excitons with large momenta. This instability can develop across the soliton profile resulting in either an imprinted stationary nanoscale periodic pattern or explosive soliton oscillations.