We analyze models where agents search for partners to form relationships (employment, marriage, etc.), and may or may not continue searching for different partners while matched. Matched agents are less inclined to search if their match yields more utility, and also if it is more stable. If one partner searches the relationship is less stable, so the other is more inclined to search, potentially making instability a self-fulfilling prophecy. We show this can generate multiple -- indeed, a continuum of -- equilibria. We investigate efficiency and show that in any equilibrium there tends to be too much turnover, unemployment, and inequality. We calibrate an example to see how well the model can account for job-to-job transitions, and to see how much endogenous instability matters.