The probabilistic serial (PS) rule is one of the most prominent randomized rules for the assignment problem. It is well-known for its superior fairness and welfare properties. However, PS is not immune to manipulative behaviour by the agents. We examine computational and non-computational aspects of strategising under the PS rule. Firstly, we study the computational complexity of an agent manipulating the PS rule. We present polynomial-time algorithms for optimal manipulation. Secondly, we show that expected utility best responses can cycle. Thirdly, we examine the existence and computation of Nash equilibrium profiles under the PS rule. We show that a pure Nash equilibrium is guaranteed to exist under the PS rule. For two agents, we identify two different types of preference profiles that are not only in Nash equilibrium but can also be computed in linear time. Finally, we conduct experiments to check the frequency of manipulability of the PS rule under different combinations of the number of agents, objects, and utility functions.