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We introduce a large family of homogeneous and isotropic cosmological solutions in quadratic gravity which are singularity-free at early and late times. This kind of smooth solutions only emerges beyond the unstable de Sitter branch $3\alpha < \beta$, $\alpha$ being the coupling of the $R^2$ term and $-\beta$ the coupling of the $R_{\mu\nu}^2$ term. We have analyzed the stability of these singularity-free solutions by computing the second-order variation of the action. The complete analysis shows that a Big Bang can emerge from a singularity-free regime when the parameters of the theory are slightly modified, revealing the unstable nature of this type of solutions., Comment: 16 pages, 16 figures