Incorporating probabilistic terms in mathematical models is crucial for capturing and quantifying uncertainties in real-world systems. However, stochastic models typically require large computational resources to produce meaningful statistics. For this reason, the development of reduction techniques becomes essential for enabling efficient and scalable simulations of complex scenarios while quantifying the underlying uncertainties. In this work, we investigate the accuracy of Polynomial Chaos (PC) surrogate expansions in capturing bifurcation phenomena, starting from generic normal forms and progressing toward a specific benchmark in fluid dynamics: the Coand\u{a} effect. In particular, we propose a novel non-deterministic approach to generic bifurcation detection problems, where the stochastic setting provides a different perspective on the non-uniqueness of the solution, also avoiding expensive simulations for many instances of the parameter. Thus, starting from the formulation of the Spectral Stochastic Finite Element Method (SSFEM), we extend the methodology to deal with solutions of a bifurcating problem, by working with a perturbed version of the deterministic PDE. We discuss the link between deterministic and stochastic bifurcation diagrams, highlighting the surprising capability of PC polynomial coefficients to give insights into the deterministic solution manifold.