We show that an Optical Parametric Oscillator which is simultaneously resonant for several modes, either spatial or temporal, generates both below and above threshold a multimode non-classical state of light consisting of squeezed vacuum states in all the non-oscillating modes. We confirm this prediction by an experiment dealing with the degenerate TEM$_{01}$ and TEM$_{10}$ modes. We show the conservation of non-classical properties when the threshold is crossed. The experiment is made possible by the implementation of a new method to lock the relative phase of the pump and the injected beam.