In liquid argon TPCs for dark matter search and neutrino detection experiments, primary scintillations are used as a prompt signal of particle scattering, being intensively produced in the vacuum ultraviolet (VUV) due to excimer emission mechanism. On the other hand, there were indications on the production of visible light scintillations in liquid argon, albeit at a much lower intensity, the origin of which is still not clear. The closely related issue is visible light scintillations in liquid argon doped with methane, the interest in which is due to the possible use in neutron veto detectors for those experiments. In this work we study in detail the properties of such scintillations in pure liquid argon and its mixtures with methane. In particular, the absolute photon yield of visible light scintillations in pure liquid argon was measured to be about 200 and 90 photon/MeV for X-rays and alpha particles respectively. In liquid argon doped with methane the photon yield dropped down significantly, by about an order of magnitude at a methane content varying from 0.01 to 1%, and then almost did not change when further increasing the methane content up to 10%. There are firm indications that the mechanism of visible light scintillations in liquid argon and its mixtures with methane is that of neutral bremsstrahlung of primary ionization electrons.