A group is defined as a set that is: associative, contains the identity element $(e)$, and every element has an inverse under given operation $(*)$. By this definition, $\{e\}$ is a group over the binary structure $(\{e\}, *)$. (I am leaving $*$ as an undefined operation for the purpose of generalizing)
Given this, is it safe to assume $\{e\}$ is cyclic because it can be generated by $\langle e \rangle$?
We could also understand this as being over the binary structure $(\mathbb{Z}_1, +_1)$ since $(\{e\}, *)$ is isomorphic to $(\mathbb{Z}_1, +_1)$.
The trivial group is indeed cyclic, if you so define it. A presentation would be $$\langle a\mid a\rangle,$$ which fits into the pattern of $$\Bbb Z_n\cong\langle x\mid x^n\rangle.$$
Sometimes, one can say that the trivial group is generated by the empty set, however, so be careful.