How well does a simple search on-the-job model fit the eighteen years of job and wage mobility of high school graduates? To answer this question we are confronted from the data with a prevalent non-compliance and exemptions from the minimum wage. We incorporate this observation in a job search model with three main ingredients: (i) search on-the-job; (ii) minimum wages, with potentially imperfect compliance or exemptions; and, (iii) exogenous wage growth on-the-job. We use panel data drawn from the NLSY79, US youth panel starting in 1979, to estimate the parameters of our simple job search model and, in particular, the extent of non-compliance/exemptions to the minimum wage. The model is solved numerically and we use simulated moments to estimate the parameters. The estimated parameters are consistent with the model and they provide a good fit for the observed levels and trends of the main job and wage mobility data. Furthermore, the estimated model indicates that the non-compliance and exemption rate with the federal minimum wage translates into a roughly 10% of jobs paying less than the minimum wage. Counterfactual experiment of increase of the compliance/non-exemption rate or the minimum wage shows a small effect on mean accepted wages but a significant negative effect on the non-employment rate.