One can classify (commutative) finite flat group schemes (with order of $p$-powers) over $\mathbb Z_p$ using semi-linear algebraic datas such as Breuil-Kisin modules. And we can fix the special fiber and form a moduli space of deformations.
How about the global story? Let $K$ be a number field, can we classify all finite flat group schemes over $O_K$ ? Can we classify finite flat group schemes over a smooth projective curve over a finite field?
The etale ones can be described in terms of Galois representations. There is an exact sequence between global Brauer group and local ones in class field theory, can we form a local-global story here similarly?
Related question: Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$ .