  The spin-curvature coupling as captured by the so-called
Mathisson-Papapetrou-Dixon (MPD) equations is the leading order effect of the
finite size of a rapidly rotating compact astrophysical object moving in a
curved background. It is also a next-to-leading order effect in the phase of
gravitational waves emitted by extreme-mass-ratio inspirals (EMRIs), which are
expected to become observable by the LISA space mission. Additionally,
exploring the Hamiltonian formalism for spinning bodies is important for the
construction of the so-called Effective-One-Body waveform models that should
eventually cover all mass ratios.
  The MPD equations require supplementary conditions determining the frame in
which the moments of the body are computed. We review various choices of these
supplementary spin conditions and their properties. Then, we give Hamiltonians
either in proper-time or coordinate-time parametrization for the
Tulczyjew-Dixon, Mathisson-Pirani, and Kyrian-Semer\'ak conditions. Finally, we
also give canonical phase-space coordinates parametrizing the spin tensor. We
demonstrate the usefulness of the canonical coordinates for symplectic
integration by constructing Poincar\'e surfaces of section for spinning bodies
moving in the equatorial plane in Schwarzschild space-time. We observe the
motion to be essentially regular for EMRI-ranges of the spin, but for larger
values the Poincar\'e surfaces of section exhibit the typical structure of a
weakly chaotic system. A possible future application of the numerical
integration method is the inclusion of spin effects in EMRIs at the precision
requirements of LISA.
