A number field $K$ is said to be monogenic when $\mathcal{O}_K=\mathbb{Z}[\alpha]$ for some $\alpha\in\mathcal{O}_K$. What is currently known about which $K$ are monogenic? Which are not? From Marcus's Number Fields, I'm familiar with the proof that the cyclotomic fields are monogenic, and for example that $\mathbb{Q}(\sqrt{7},\sqrt{10})$ is not monogenic (it is exercise 30 of chapter 2), but because Marcus eschews anything local, I haven't seen any of the perhaps more natural proofs of these results.

If $K$ is monogenic, is there an effective method of determining those $\alpha\in\mathcal{O}_K$ for which $\mathcal{O}_K=\mathbb{Z}[\alpha]$?

More generally, what is known about the minimal number of generators of $\mathcal{O}_K$ as a $\mathbb{Z}$-algebra? That is, can we determine, or at least put non-trivial bounds on, the minimal $m$ such that $\mathcal{O}_K=\mathbb{Z}[\alpha_1,\ldots,\alpha_m]$ for some $\alpha_i\in\mathcal{O}_K$? We know that any $\mathcal{O}_K$ has an integral basis of $n=[K:\mathbb{Q}]$ elements, so certainly $m\leq n$ (I'm considering that trivial).