Galaxies spin so fast that the matter we can see lit up should be unable to hold them in by gravity and they should explode. Oddly they don't, so astronomers have speculated that dark (invisible) matter exists in the galaxies to hold them in with an extra gravitational force. This hypothesis has become entrenched in the astrophysics community to the exclusion of all other hypotheses despite a lack of evidence after decades of searching. This is perhaps because of its infinite flexibility which makes it difficult to disprove, so here I'd like to discuss some evidence against dark matter.
Dark matter usually only needs to be added to the edges of galaxies since in their centres they behave normally. Sanders and McGaugh (2002) pointed out that the radius at which galaxies start to spin too fast for their own good, and to need dark matter, is not a set distance, but it always occurs where the rotational acceleration drops below 1.2*10^-10 m/s^2: a very low acceleration called 'a0': a regime not previous encountered by our experiments. This is difficult to explain by dark matter - you'd have to invent a kind of matter that suddenly appears when accelerations are below this value.
Since dark matter is needed only at the galactic edge, its supporters need to have some new physics that keeps it smooth and diffuse. Brilliantly poking a hole in that, Scarpa et al. (2006) looked at globular clusters which are small dense congregations of stars within the galaxy, a bit like clumps of mistletoe in an oak tree. They found that whenever the 'internal' acceleration of these clusters drops below 1.2*10^-10 m/s^2 (a0 again) they spin far too fast to be stable, just like the full-sized galaxies, but in these globular clusters this anomaly cannot be explained by dark matter, since to work for galaxies dark matter must be smooth on these smaller scales.
An empirical hypothesis suggested by Mordehai Milgrom to explain galaxy rotation is called MoND (Modified Newtonian Dynamics). MoND doesn't have a physical model, but says that when total accelerations are below the critical acceleration a0 then either the gravitational mass of stars goes up, or their inertial mass goes down (in MoND you can choose either). MoND explains disc galaxies well, but it cannot explain these globular clusters because, as Scarpa et al. say: the external gravity field due to the Milky Way acting on these clusters is above a0 so MoND behaviour should not appear.
MiHsC has a better chance of explaining these clusters because in MiHsC the inertia of a star depends on the mutual accelerations between the star and all other stars, but the closer stars in the cluster have more weight in the calculation, so the crucial factor determining behaviour will be the internal accelerations, as observed. I need now to work out how to model these clusters with MiHsC. For modelling galaxies with MiHsC, see the paper: McCulloch (2012), or a brief summary.
An even better crucial test (simpler to model) would be to use the smallest globular clusters of all: wide binaries. Some binary stars with wide orbits have accelerations below a0, and they also seem to show anomalous behaviour (see Hernandez et al., 2011) but the data is noisy and not yet conclusive. Note: an even better test is the Alpha Centauri system.
Sanders. R.H., and S.S. McGaugh, 2002. Modified Newtonian Dynamics as an alternative to dark matter. Ann. Rev. Astro. and Astrop., 40, 263. Preprint. http://arxiv.org/astro-ph/0204521
Scarpa, R., G. Marconi, R. Gilmozzi, 2006. Globular clusters as a test for gravity in the weak acceleration regime. Proceedings of the 1st crisis in cosmology conference. Am. Inst. Phys Proceedings series, Vol. 822. Preprint. http://arxiv.org/astro-ph/0601581
McCulloch, M.E., 2012. Testing quantised inertia on galactic scales. Astrophysics and Space Science, Vol. 342, No. 2, 575-578. Preprint. http://arxiv.org/abs/1207.7007
Hernandez, X., M.A. Jimenez and C. Allen, 2011. Wide binaries as a critical test of classical gravity. Euro. Phys. J. C., 72, 1884. Preprint. http://arxiv.org/abs/1105.1873