<p>In this work, a generalized stochastic scale-invariant analogue of the Korteweg-de Vries (SIdV) equation is investigated. By taking the stochastic wave transformation, the considered model is transformed into the reduced form. And the Gaussian soliton solutions under the effect of noise are derived. Then we convert the reduced model into the planar dynamical system, various phase portraits under specific conditions are obtained and analyzed via qualitative theory. Subsequently, we employ the complete discrimination system for polynomial method to construct a variety of traveling wave solutions. In particular, the peakon soliton solutions and singular solutions under the effect of noise are presented graphically. Finally, we perform a detailed discussion on the chaotic behaviors of this system by introducing the external perturbation terms. The results such as Gaussian soliton solutions, qualitative properties and chaotic behaviors for the generalized stochastic SIdV equation are initially investigated in the present paper.</p>