We study the McKean-Vlasov equation
∂t̺ = β−1∆̺ + κ ∇·(̺∇(W ⋆ ̺)) ,
with periodic boundary conditions on the torus. We first study the global asymptotic stability of the
homogeneous steady state. We then focus our attention on the stationary system, and prove the existence
of nontrivial solutions branching from the homogeneous steady state, through possibly infinitely many
bifurcations, under appropriate assumptions on the interaction potential. We also provide sufficient
conditions for the existence of continuous and discontinuous phase transitions. Finally, we showcase
these results by applying them to several examples of interaction potentials such as the noisy Kuramoto
model for synchronisation, the Keller–Segel model for bacterial chemotaxis, and the noisy Hegselmann–
Krausse model for opinion dynamics.