This paper reviews four variants of global Carleman weights that are especially adapted to some singular controllability and inverse problems in partial differential equations. These variants arise when studying: i) one measurement stationary source inverse problems for the heat equation with discontinuous coefficients, ii) one measurement stationary potential inverse problems for the heat equation with discontinuous coefficients, iii) null controllability for fluid-structure problems in mobile domains and iv) recovering coefficients from locally supported boundary observations for the wave equation. In all the case we explain how to explicitly construct the Carleman weight.