Exclusive production of omega mesons was studied at the COMPASS experiment by scattering 160GeV/c muons off transversely polarised protons. Five single-spin and three double-spin azimuthal asymmetries were measured in the range of photon virtuality 1(GeV/c)(2) < Q(2) < 10(GeV/c)(2), Bjorken scaling variable 0.003 < xBj < 0.3 and transverse momentum squared of the omega meson 0.05(GeV/c)(2) < p(T)(2) < 0.5(GeV/c)(2). The measured asymmetries are sensitive to the nucleon helicity-flip Generalised Parton Distributions (GPD) Et hat are related to the orbital angular momentum of quarks, the chiral-odd GPDs H-T that are related to the transversity Parton Distribution Functions, and the sign of the pi omega transition form factor. The results are compared to recent calculations of a GPD-based model. (C) 2016 The Author(s). Published by Elsevier B.V.