We investigate ${\mathcal F}$-hypercyclicity of linear, not necessarily continuous, operators on Fréchet spaces. The notion of lower $(m_n)$-hypercyclicity seems to be new and not considered elsewhere even for linear continuous operators acting on Fréchet spaces. We pay special attention to the study of $q$-frequent hypercyclicity, where $q\geq 1$ 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.