Functional determinants resulting from functional integration in quantum gauge theories are studied. We derive an expansion around the constant field strength for the (renormalized) spinor determinant ${\mathrm{det}\mathcal{M}}_{F}$ in QED. We show that, if the field strength $F$ is large and its derivatives are bounded, then ${\mathrm{det}\mathcal{M}}_{F}\ensuremath{\equiv}\mathrm{exp}(\ensuremath{-}W)\ensuremath{\sim}\mathrm{exp}(c\ensuremath{\int}{F}^{2}\mathrm{ln}{F}^{2})$, where $c>0$. Hence, the effective action $W$ in (four-dimensional) QED is unbounded from below. Moreover, we prove that $\mathrm{exp}(\ensuremath{-}W)$ is not integrable. A similar result is established in the Yukawa model [${\mathrm{det}\mathcal{M}}_{Y}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\int}{\ensuremath{\varphi}}^{4}\mathrm{ln}{\ensuremath{\varphi}}^{4})$]. We estimate the scalar determinant ${{\mathrm{det}\mathcal{M}}_{A}}^{2}$ for a non-Abelian gauge field. We show that (like in the Abelian case studied earlier) ${{\mathrm{det}\mathcal{M}}_{A}}^{2}=\mathrm{exp}[c\ensuremath{\int}{|F|}^{2}\mathrm{ln}{|F|}^{2}+{r}_{2}(F,DF,DDF)]$, where $c>0$ and ${r}_{2}$ is bounded by a quadratic form of the gauge-invariant variables $|F|$, $|\mathrm{DF}|$, and $|\mathrm{DDF}|$. We investigate the effect of gluon self-interaction on the stability of models with broken gauge symmetry $G\ensuremath{\rightarrow}H$ (we discuss in detail the Georgi-Glashow model). We sum up (in an approximation) the contribution of massive gluons to the O(2)-invariant effective action. It is shown that this effective action is bounded from below for slowly varying fields, if the couplings are asymptotically free at the one-loop level.