Watch the Melody: Stress and Stabillity online guitar lesson by Matthieu Brandt from Songwriting on Guitar
Stable melody notes are notes that can be found in the background chord. By targeting these notes you'll flow with the underlying harmony. Non chord tones will create some tension and make the melody flow forward.
Depending on how long you play/sing these notes, on which location in the bar and how loud you play them, the resulting effect will create more or less stress or stabillity. A large pitch shift between a note and its preceding note will underline this effect as does the use of syncopation and anticipation.