Membrane remodelling processes involving the formation and fission of small buds require the formation and closure of narrow membrane necks, both for biological membranes and for model membranes such as lipid bilayers. The conditions required for the stability of such necks are well understood in the context of budding of vesicles with bilayer asymmetry and/or intramembrane domains. In many cases, however, the necks form in the presence of an adhesive surface, such as a solid particle or substrate, or the cellular cortex itself. Examples of such processes in biological cells include endocytosis, exocytosis and phagocytosis of solid particles, the formation of extracellular and outer membrane vesicles by eukaryotic and prokaryotic cells, as well as the closure of the cleavage furrow in cytokinesis. Here, we study the interplay of curvature elasticity, membrane-substrate adhesion, and constriction forces to obtain generalized stability conditions for closed necks which we validate by numerical energy minimization. We then explore the consequences of these stability conditions in several experimentally accessible systems such as particle-filled membrane tubes, supported lipid bilayers, giant plasma membrane vesicles, bacterial outer membrane vesicles, and contractile rings around necks. At the end, we introduce an intrinsic engulfment force that directly describes the interplay between curvature elasticity and membrane-substrate adhesion.