A signed graph $\Gamma$ is a graph whose edges are labeled by signs. If $\Gamma$ has $n$ vertices, its spectral radius is the number $\rho \left(\Gamma \right):=max\left\{\right|{\lambda }_i\left(\Gamma \right)|:1\le i\le n\}$, where ${\lambda }_1\left(\Gamma \right)\ge \cdots \ge {\lambda }_n\left(\Gamma \right)$ are the eigenvalues of the signed adjacency matrix $A\left(\Gamma \right)$. Here we determine the signed graphs achieving the minimal or the maximal spectral radius in the classes ${𝔘}_n$ and ${𝔅}_n$ of unbalanced unicyclic graphs and unbalanced bicyclic graphs, respectively.