Phase transitions are widespread phenomena occurring during the early stages of the Universe as well as in our everyday experience. Many systems enjoy similar critical properties governed by symmetries, range of interaction, and dimensionality, disregarding microscopic properties. At the same time, some of them are much easier to recreate in the lab than others. A particular boost in the investigation of phase transitions is associated with the development of quantum simulators, such as light-matter interfaces, trapped ions, Rydberg atoms, etc., which allow to controllably study critical properties at near-zero or finite temperatures. Such simulators allow for tuning of parameters, which can be quite robust in the real-world system, driving transition dynamically as well as giving access to real-time observables. It is also important that these platforms enable not only coherent control of couplings but also can host incoherent, dissipative channels due to the environment in which they are embedded. Over the last years, such dissipations have been shown to be more than just a source of decoherence, but rather a resource for more sophisticated control of dynamics, allowing the generalization of new phases or stabilizing dynamical regimes. In this thesis, we study phase transitions in the experimentally accessible analogous quantum simulators. We start with a spintronics-inspired dissipative model, which hosts various phases generated through the competition of continuous and discrete symmetries and incoherent drive and dissipations. The model possesses a rich dynamical phase diagram, combining stationary and non-stationary regimes. Then, we go deeper into the experimental implementation of the models with different atomic degrees of freedom and see how the interplay of spin and momentum can result in non-stationary phases, manifesting spin-momentum entanglement. We also examine a particular role of dissipation in this case, showing how it helps stabilize dynamical phases and gives a perspective for non-destructive monitoring of the system properties. The thesis covers the implementation of dissipative models in quantum simulators and their relevant dynamical aspects.