  The focus is on circular nets with one or two families of spherical parameter
lines, which are treated in M\"obius geometry. These circular nets provide a
discretisation of surfaces with one or two families of spherical curvature
lines. The special cases of planar, circular and linear parameter lines are
also investigated. A Lie-geometric discretisation in terms of principal contact
element nets is also presented. Its properties are analogous to the classical
properties of surfaces with one or two families of spherical curvature lines.
Circular nets with two families of spherical parameter lines have geometric
properties that are related to Darboux cyclides. Circular nets with one or two
families of spherical parameter lines are examples of Q-nets with terminating
Laplace sequences. More generally, this article considers Q-nets that are
inscribed in quadrics and that have terminating Laplace sequences.
