  We study surfaces with one constant principal curvature in Riemannian and
Lorentzian three-dimensional space forms. Away from umbilic points they are
characterized as one-parameter foliations by curves of constant curvature, each
of these curves being centered at a point of a regular curve and contained in
its normal plane. In some cases, a kind of trichotomy phenomenon is observed:
the curves of the foliations may be circles, hyperbolas or horocycles,
depending of whether the constant principal curvature is respectively larger,
smaller of equal to one (not necessarily in this order). We describe explicitly
some examples showing that there do exist complete surfaces with one constant
principal curvature enjoying both umbilic and non-umbilic points.
