We investigate the dynamical features of a large family of running vacuum cosmologies for which evolves as a polynomial in the Hubble parameter. Specifically, using the critical point analysis we study the existence and the stability of singular solutions which describe de-Sitter, radiation and matter dominated eras. We find several classes of (H) cosmologies for which new analytical solutions are given in terms of Laurent expansions. Finally, we show that the Milne universe and the R_h = ct model can be seen as perturbations around a specific (H) model, but this model is unstable.