We investigate F-hypercyclicity of linear, not necessarily continuous, operators on Frechet spaces. The notion of lower (mn)-hypercyclicity seems to be new and not considered elsewhere even for linear continuous operators acting on Frechet spaces. We pay special attention to the study of q-frequent hypercyclicity, where q > 1 is an arbitrary real number. We present several new concepts and results for lower and upper densities in a separate section, providing also a great number of illustrative examples and open problems.