In this paper we describe a family of compatible Poisson structures defined on the space of coframes (or differential invariants) of curves in flat homogeneous spaces of the form
M
≅
(
G
⋉
R
n
)
/
G
\mathcal {M} \cong (G\ltimes \mathbb {R}^n)/G
where
G
⊂
G
L
(
n
,
R
)
G\subset {\mathrm {GL}}(n,\mathbb {R})
is semisimple. This includes Euclidean, affine, special affine, Lorentz, and symplectic geometries. We also give conditions on geometric evolutions of curves in the manifold
M
\mathcal {M}
so that the induced evolution on their differential invariants is Hamiltonian with respect to our main Hamiltonian bracket.