It is false that a flat Levi-Civita connection on a manifold necessarily has zero holonomy group. The theorem of Bieberbach - I think - is that the holonomy group must be finite.
For instance,the flat Klein bottle has holonomy Z2.
Here is a 3 dimensional flat orientable Riemannian manifold with holonomy group,Z4.
Take the quotient of R^3 with the standard flat metric by the group of isometries generated by the standard lattice( vectors all of whose coordinates are integers) together with the isometry (x,y,z) -> (x +1/4, z,-y)
It is probably true that any finite group can be the holonomy group of a flat Riemannian manifold.
The classifying theorem is that any compact flat Riemannian manifold without boundary is the quotient of R^n by a torsion free group that satisfies an exact sequence
o -> L -> G ->G/L -> 1
where L is a full lattice and G/L is a finite group. G/L is isomorphic to the holonomy group of the quotient flat manifold. The converse is also true. Any such group is the fundamental group of a flat Riemannian manifold.
There are flat Riemannian manifolds in every dimension as can be seen by taking the Cartesian product of the Klein bottle with flat tori.
I believe that there is no known method for identifying all of the flat manifolds. There may be classifications in low dimensions such as dimension 3 or 4. In dimension 2 the only ones are the flat torus and the flat Klein bottle.
The overwhelming difficulty in classification is that there is no way to classify the inequivalent integral representations of finite groups. Even very small groups - with only several exceptions such as Z2- have infinitely many inequivalent Z representations. This comes up because the quotient group G/L acts on the lattice,L, by conjugation (since the lattice is a normal subgroup of G) and this action is an integral representation. I say this just to illustrate that the condition of a flat Levi - Civita connection is extremely complex.
Even though the Pontryagin classes and Euler class of a flat Riemannian manifold must be zero - by Chern-Weil theory - the Stiefel- Whitney classes do not.For instance the first Stiefel-Whitney class of the Klein bottle is not zero. I would be surprised if there is a theorem restricting the possible Stiefel Whitney classes except that the top class must be zero.
Finally I would point out that the Moebius band made out of a piece of paper is a flat manifold with boundary and its holonomy group is Z2.