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Math. Anal."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We study the nonhomogeneous Dirichlet problem for first-order Hamilton--Jacobi equations associated with Tonelli Hamiltonians on a bounded domain $\\Omega$ of $\\mathbb{R}^n$ assuming the energy level to be supercritical. First, we show that the viscosity (weak KAM) solution of such a problem is Lipschitz continuous and locally semiconcave in $\\Omega$. Then, we analyze the singular set of a solution showing that singularities propagate along suitable curves, the so-called generalized characteristics, and that such curves stay singular unless they reach the boundary of $\\Omega$. Moreover, we prove that the latter is never the case for mechanical systems and that singular generalized characteristics converge to a critical point of the solution in finite or infinite time. 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