In a cylindrical pore of arbitrary wettability, we analyse the existence of a three-fluid contact line connecting the fluid-fluid interfaces between two bulk phases and the third phase contained in a cusp near the pore wall. This analysis is Supported by the very similar, but simpler, analysis for a constriction between parallel plates. From the force balance at the contact line and the equations for the interface curvatures we derive expressions for the cusp height and for the capillary entry pressure related to piston-like displacement between the two bulk phases. The latter is independent of the existence of a cusp and its phase pressure. Based on some realistic assumptions, of which the most important is that a cusp grows continuously from the onset when its phase pressure is increased, we analyse under which conditions a cusp can exist, and, when it exists, what its behaviour is as a function of the cusp phase pressure. We find a simple criterion involving (two ratios of) the three interfacial tensions and two of the three contact angles, which determines whether the three-fluid contact line and, consequently, a cusp exists. The range of contact angles, as well its the size of the cusp increases, when the cusp phase is close to spreading. Not only cusps of the wetting phase can occur, but also of the intermediate-wetting phase. Numerical examples are presented to illustrate the range of behaviour of the cusps. (c) 2005 Elsevier Inc. All rights reserved.