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In this paper, under very general assumptions, we prove existence and regularity of distributional solutions to homogeneous Dirichlet problems of the form $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle - \Delta _{1} u = h(u)f &{} \text {in}\, \Omega , \\ u\ge 0&{} \text {in}\ \Omega ,\\ u=0 &{} \text {on}\ \partial \Omega \,. \end{array}\right. } \end{aligned}$$ Here $$\Delta _{1} $$ is the 1-Laplace operator, $$\Omega $$ is a bounded open subset of $$\mathbb {R}^N$$ with Lipschitz boundary, h(s) is a continuous function which may become singular at $$s=0^{+}$$ , and f is a nonnegative datum in $$L^{N,\infty }(\Omega )$$ with suitable small norm. Uniqueness of solutions is also shown provided h is decreasing and $$f>0$$ . As a preparatory tool for our method a general theory for the same problem involving the p-Laplacian (with $$p>1$$ ) as principal part is also established. The main assumptions are further discussed in order to show their optimality.