We focus on reduced order modelling for nonlinear parametrized Partial Differential Equations, frequently used in the mathematical modelling of physical systems. A common issue in this kind of problems is the possible loss of uniqueness of the solution as the parameters are varied and a singular point is encountered. In the present work, the numerical detection of singular points is performed online through a Reduced Basis Method, coupled with a Spectral Element Method for the numerically intensive offline computations. Numerical results for laminar fluid mechanics problems will be presented, where pitchfork, hysteresis, and Hopf bifurcation points are detected by an inexpensive reduced model. Some of the presented 2D and 3D flow results deal with the study of instabilities in a simplified model of a mitral regurgitant flow in order to understand the onset of the Coanda effect. The first results are in good agreement with the reference.