Bright circumstellar nebulae around massive stars are potentially useful to derive time-dependent mass-loss rates and hence constrain the evolution of the central stars. A key case in this context is the relatively young ejection-type nebula M1-67 around the runaway Population I Wolf-Rayet star WR 124 (=209 BAC), which exhibits a WN 8 spectrum. With HST-WFPC2 we have obtained a deep, Hα image of M1-67. This image shows a wealth of complex detail that was briefly presented previously by Grosdidier et al. With the interferometer of the Université Laval (Québec, Canada), we have obtained complementary Fabry-Perot Hα data using Canada-France-Hawaii Telescope (CFHT) MOS/SIS. From these data M1-67 appears more or less as a spherical (or elliptical, with the major axis along the line of sight), thick, shell seen almost exactly along its direction of rapid spatial motion away from the observer in the ISM. However, a simple thick shell by itself would not explain the observed multiple radial velocities along the line of sight. This velocity dispersion leads one to consider M1-67 as a thick accelerating shell. Given the extreme perturbations of the velocity field in M1-67, it is virtually impossible to measure any systematic impact of the present WR (or previous LBV) wind on the nebular structure. The irregular nature of the velocity field is likely due to either large variations in the density distribution of the ambient ISM or large variations in the central star mass-loss history. In addition, either from the density field or the velocity field, we find no clear evidence for a bipolar outflow, as was claimed in other studies. On the deep Hα image we have performed continuous wavelet transforms to isolate stochastic structures of different characteristic size and look for scaling laws. Small-scale wavelet coefficients show that the density field of M1-67 is remarkably structured in chaotically (or possibly radially) oriented filaments everywhere in the nebula. We draw attention to a short, marginally inertial range at the smallest scales (6.7-15.0×10-3 pc), which can be attributed to turbulence in the nebula, and a strong scale break at larger scales. Examination of the structure functions for different orders shows that the turbulent regime may be intermittent. Using our Fabry-Perot interferograms, we also present an investigation of the statistical properties of fluctuating gas motions using structure functions traced by Hα emission-line centroid velocities. We find that there is a clear correlation at scales 0.02-0.22 pc between the mean quadratic differences of radial velocities and distance over the surface of the nebula. This implies that the velocity field shows an inertial range likely related to turbulence, though not coincident with the small inertial range detected from the density field. The first- and second-order moments of the velocity increments are found to scale as <|Δv(r)|>~r0.5 and <|Δv(r)|2>~r0.9. The former scaling law strongly suggests that supersonic, compressible turbulence is at play in the nebula; on the other hand, the latter scaling law agrees very well with Larson-type laws for velocity turbulence. Examination of the structure functions for different orders shows that the turbulent regime is slightly intermittent and highly multifractal with universal multifractal indexes α~1.90-1.92 and C1~0.04+/-0.01. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. Also based on observations collected at the Canada-France-Hawaii Telescope (CFHT), which is operated by CNRS of France, NRC of Canada, and the University of Hawaii.