We present sublinear-time (randomized) algorithms for finding simple cycles of length at least $k\geq 3$ and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being $C_k$-minor-free (resp., free from having the corresponding tree-minor). In particular, if the graph is far (i.e., $Ω(1)$-far) {from} being cycle-free, i.e. if one has to delete a constant fraction of edges to make it cycle-free, then the algorithm finds a cycle of polylogarithmic length in time $\tildeO(\sqrt{N})$, where $N$ denotes the number of vertices. This time complexity is optimal up to polylogarithmic factors. The foregoing results are the outcome of our study of the complexity of {\em one-sided error} property testing algorithms in the bounded-degree graphs model. For example, we show that cycle-freeness of $N$-vertex graphs can be tested with one-sided error within time complexity $\tildeO(\poly(1/\e)\cdot\sqrt{N})$. This matches the known $Ω(\sqrt{N})$ query lower bound, and contrasts with the fact that any minor-free property admits a {\em two-sided error} tester of query complexity that only depends on the proximity parameter $\e$. For any constant $k\geq3$, we extend this result to testing whether the input graph has a simple cycle of length at least $k$. On the other hand, for any fixed tree $T$, we show that $T$-minor-freeness has a one-sided error tester of query complexity that only depends on the proximity parameter $\e$. Our algorithm for finding cycles in bounded-degree graphs extends to general graphs, where distances are measured with respect to the actual number of edges. Such an extension is not possible with respect to finding tree-minors in $o(\sqrt{N})$ complexity.