I will present some results related with existence and sharp regularity for solutions to a class of singular elliptic equations with gradient dependence for which solutions may exhibit a free boundary. Once we have obtained sharp regularity, further analysis of the free boundary may be carried out with nondegeneracy and refined gradient estimates. I will also show how to employ an asymptotic analysis obtaining some known results for the obstacle and cavity problems. Work done in collaboration with M. Montenegro and E. Teixeira.