Have $p$-groups with a unique normal minimal subgroup been classified? Is there any article on the subject?

As Todd Leason said, a $p$-group has a unique minimal normal subgroup if and only if it has cyclic center. Here is some evidence that there is no reasonable classification of such groups:

The class of these groups is big: It contains the upper unipotent groups over prime fields, as YCor said, it contains the Sylow $p$-subgroups of the symmetric group $S_{p^n}$, the $p$-groups of maximal nilpotency class, the extraspecial groups and lots of other groups.

Every $p$-group embeds into such a group (follows from each of the first two examples in 1.).

Every $p$-group $P$ has factor groups of this type, and at least one factor group with the same nilpotency class as $P$: Namely, if $D\colon P \to \operatorname{GL}(d,\mathbb{C})$ is an irreducible representation, then $P/\operatorname{Ker}(D)$ has cyclic center, and there must be an irrep such that its kernel does not contain the last non-trivial term, $P^c$, of the decending central series of $P$.

andcyclic commutator subgroup. But I don't know if there is a classification of p-groups with cyclic center. $\endgroup$ – Todd Leason Feb 9 '16 at 21:43