Now let's see how a singular point works. There are many perspectives on this, and the Applet allows you to see them all. If we follow along a stable coincidence curve in P-T space we will always have two (X) curves that touch at a single point, and one must be "fatter" than the other. As the (X) curve of a third phase moves around, the point of tangency where the fatter of the two coincident phases and the third phase are in equilibrium may shift so that it becomes equal to the coincident point. At this location, the singular point, all three phases are in mutual equilibrium. If we keep going in the same direction, the (X) curve of the third phase will keep shifting in the same sense and it is necessary that the common tangent between the third phase and the fatter of the two coincident phases come down below the coincident point and turn it into a metastable coincidence.
A second perspective on a singular point is seen by following along a three-phase univariant curve; in general the three phases each have distinct compositions where they coexist. However, as we move along the curve the coexisting compositions can shift and it may happen that two of them approach each other. At a certain point, two of the compositions coexisting along the univariant cross each other, so that one coefficient in the stoichiometry of the univariant reaction changes sign (i.e., A+B=Liquid changes to A+Liquid=B). Right at the point where the compositions cross, they are equal, and evidently we are also on the coincidence curve at this point. Hence this special point is the same singular point we saw before, we have just crossed it in a different direction. The three-phase univariant is stable on both sides, it just changes stoichiometry. If the phase that changes sign is a liquid, then a singular point is where a peritectic changes into a eutectic or vice versa.
Since this applet was written, browers have stopped supporting applets for security reasons. You can download a standalone executable that runs all the functionality of the applet on your own platform here. When you run it, select "Example 7" to get this page's content.