  With the aid of concrete examples, we consider the question of whether, in
the presence of conformal curvature, a conformal geodesic can become trapped in
smaller and smaller sets, or phrased informally: are spirals possible? We do
not arrive at a definitive answer, but we are able to find situations where
this behaviour is ruled out, including a reduction of the conformal-geodesic
equations to quadratures in a specific non-conformally flat metric.
