  We study the stability/instability of the subsonic travelling waves of the
Nonlinear Schr\"odinger Equation in dimension one. Our aim is to propose
several methods for showing instability (use of the Grillakis-Shatah-Strauss
theory, proof of existence of an unstable eigenvalue via an Evans function) or
stability. For the later, we show how to construct in a systematic way a
Liapounov functional for which the travelling wave is a local minimizer. These
approaches allow to give a complete stability/instability analysis in the
energy space including the critical case of the kink solution. We also treat
the case of a cusp in the energy-momentum diagram.
