This seminar addresses the existence and optimal global regularity of solutions for homogeneous Dirichlet problems involving the 1-Laplacian operator and singular lower-order terms. The motivation for studying such equations stems from several applied models, including the Blasius model for non-Newtonian boundary layers in fluid dynamics, perfect plastic torsion models without viscosity, and Total Variation regularization in image processing. Historically, for the standard quasi-linear operator, it is known that there exists a critical threshold for the singularity exponent above which solutions lose global finite energy. However, by taking the limit as the operator approaches the 1-Laplacian, this threshold formally tends to infinity. This observation leads to the natural conjecture that in the limiting 1-Laplacian case, the singularity cannot destroy the finite energy property. In this talk, we will then present a recent result establishing that, under suitable integrability conditions on the data, unique (when expetcted) positive solutions exist and belong to the space $BV(\Omega)$, even in the presence of strong singularities. Finally, we will detail the proof strategy, focusing on truncation techniques, the splitting method, and the analysis of the energetic cost and the tail. This is a join twork in collaboration with Antonio Jesús Martínez Aparicio (Univ. of Almeria, Spain) and Francescantonio Oliva (Sapienza, University of Rome). This seminar is part of the activities of the Excellence Department Project CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.