  We investigate the dynamical features of a large family of running vacuum
cosmologies for which $\Lambda$ 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 $\Lambda(H)$
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 $\Lambda(H)$ model, but this
model is unstable.
