If the students concentrate on a particular portion of the slinky, they will see that that part goes up and down as the wave passes perpendicular to it. You can also illustrate the concept of shear deformation with the slinky in its original, unstretched shape. Ordinarily it has the shape of a cylinder, or if viewed from the side, a rectangle. Holding this cylinder vertically, with one hand on the top and one on the bottom, move the top part horizontally. The individual coils of the slinky will move progressively to the side, much like a deck of cards.
Shear deformation involves a change of shape with no volume change.
Since the slinky is really just a coiled wire, both P and S waves in the slinky occur by bending these coils. Thus, the incompressibility and rigidity of the slinky are directly related to the resistance of these coils being bent. This means that, unlike the solid materials of the Earth, P waves and S waves in a slinky will probably travel at similar speeds. You don't have to mention this for the purposes of the demonstration, though.