  Cosmological models with time dependent $\Lambda$ (read as $\Lambda (t)$)
have been investigated widely in the literature. Models that solve background
dynamics analytically, are of special interest. Additionally, the allowance of
past or future singularities at finite cosmic time in a specific model signals
for a generic test on its viabilities with the current observations. Following
these, in this work we consider a variety of $\Lambda (t)$ models focusing on
their evolutions and singular behaviour. We found that a series of models in
this class can be exactly solved when the background universe is described by a
spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) line element.
The solutions in terms of the scale factor of the FLRW universe offer different
universe models, such as power law expansion, oscillating, and the singularity
free universe as well. However, we also noticed that a large number of models
in this series permits past or future cosmological singularities at finite
cosmic time. At last we close the work with a note that the avoidance of future
singularities is possible for certain models under some specific restrictions.
