We investigate the signals of a supersymmetric scenario where the lighter tausneutrino is the lightest supersymmetric particle, while the lighter stau-state is the next lightest. We confirm that such a scenario can be motivated within the framework of minimal supergravity, with just the addition of a right-chiral neutrino superfield. Such a spectrum leads to rather unusual signals of supersymmetry, showing stable tracks of the stau in the muon chambers. We have studied two types of signals, namely, (a) two or more hard jets and two stau tracks, and (b) hard jets with two muons and two stau tracks. We demonstrate that the stau tracks can be distinguished from the muonic ones through proper kinematic cuts which also enable one to remove all standard model backgrounds.