View AbstractAbstract: We will discuss three situations in which pseudoconcave sets arise as obstacles to construction of strictly plurisubharmonic (psh) functions of some class. 1. Minimal kernels of weakly complete manifolds are smallest subsets in the complement of which a continuous (or smooth) psh exhaustion function can be made strictly psh. Breaking the kernel into the union of compact pseudoconcave sets shows that a weakly complete manifold is Stein iff it does not contain a compact pseudoconcave set. (Z.S. & G.Tomassini, 2004) 2. The core of a relatively pseudoconvex domain, the largest set on which the Levi form of every smooth psh defining function of the domain must be degenerate everywhere, was introduced and shown pseudoconcave by Shcherbina et al (2016/17). We prove their conjecture that the core can be decomposed into the union of pseudoconcave sets on which every psh defining function is constant. 3. Analogous phenomena will be exhibited in relation to Richberg's regularization of strongly psh functions on complex manifolds.