Cultural trends and popularity cycles can be observed all around us, yet our theories of social influence and identity expression do not explain what perpetuates these complex, often unpredictable social dynamics. We propose a theory of social identity expression based on the opposing, but not mutually exclusive, motives to conform and to be unique among one's neighbors in a social network. We then model the social dynamics that arise from these motives. We find that the dynamics typically enter random walks or stochastic limit cycles rather than converging to a static equilibrium. We also prove that without social network structure or, alternatively, without the uniqueness motive, reasonable adaptive dynamics would necessarily converge to equilibrium. Thus, we show that nuanced psychological assumptions (recognizing preferences for uniqueness along with conformity) and realistic social network structure are both necessary for explaining how complex, unpredictable cultural trends emerge.