There are several data sets that suggest the presence of Dark Matter:
1) Orbital velocities of galaxies within a cluster are too high -- the galaxies should fly apart unless much more mass is present.
2) Observed rotational velocities of edge-on galaxies are wrong: stars near the edge rotate too fast -- unless there's a cloud of mass beyond the observed disk.
3) Gravitational lensing effects are too strong for the observed mass of the lensing clusters.
4) Numerical simulations modeling the Big Bang up to present times work well only if Dark Matter is assumed.
There may be other data sets, but these are the ones that were presented in the Coursera class I took last year titled, "Galaxies and Cosmology".