Figure skating is a beautiful sport combining elegance, precision, and athleticism. To understand some of the mechanics and complexity involved in this sport, we derive and analyze a three-dimensional model of a figure skater in continuous contact with the ice (i.e., no jumps). We model the skater as a three-dimensional body moving in space subject to a non-holonomic constraint enforcing movement along the skate's direction and holonomic constraints of continuous contact with ice and pitch constancy of the skate. We derive a surprising result that for a static (i.e., non-articulated) skater, the system is integrable if and only if the projection of the center of mass on the skate's direction coincides with the contact point with ice and some mild (and realistic) assumptions on the directions of inertia's axes. We also consider the case when the projection of the center of mass on the skate's direction does not coincide with the contact point and show that this non-integrable case exhibits apparent chaotic behavior. After a pedagogical introduction to Hamel's approach to mechanics, we extend the study to see how an articulated skater traces trajectories on ice by considering a two-dimensional skater (a Chaplygin's sleigh) with a controlled moving mass. We derive a control procedure by approximating the trajectories using circular arcs. We show that there is a control procedure minimizing the 'relative kinetic energy' of the control mass, which leads to well-posed equations for the control masses. We demonstrate examples of our system tracing actual compulsory figure skating trajectories. We also discuss further extensions of the model and applications to real-life figure skating.