We characterize a cosmic rest frame in which the monopole variation of the spherically averaged nonlinear Hubble expansion is most uniform, under arbitrary local Lorentz boosts of the central observer. Using the COMPOSITE sample of 4534 galaxies, we identify a degenerate set of candidate minimum nonlinear variation frames, which includes the rest frame of the Local Group (LG) of galaxies, but excludes the standard Cosmic Microwave Background (CMB) frame. Candidate rest frames defined by a boost from the LG frame close to the plane of the galaxy have a statistical likelihood similar to the LG frame. This may result from a lack of constraining data in the Zone of Avoidance. We extend our analysis to the Cosmicflows-2 (CF2) sample of 8162 galaxies. While the signature of a systematic boost offset between the CMB and LG frame averages is still detected, the spherically averaged nonlinear expansion variation in all rest frames is significantly larger in the CF2 sample than would be reasonably expected. We trace this to the CF2 distances being reported without a correction for inhomogeneous distribution Malmquist bias. Systematic differences in the inclusion of the large SFI++ subsample into the COMPOSITE and CF2 catalogues are analysed. Our results highlight the importance of a careful treatment of Malmquist biases for future peculiar velocities studies, including tests of the hypothesis of Wiltshire et al [Phys. Rev. D 88, 083529 (2013)] that a significant fraction of the CMB temperature dipole may be nonkinematic in origin.