Notice the characteristic ripples on the icicle--see Ripples on Icicles for videos showing how these ripples form. Visit Ice, Ice, Baby! and Geomefreeze to find out more about how icicles grow. To learn about the physics of freezing and melting water, see the Hyperphysics page Phase Changes.
credit: Stephen Morris, The Experimental Nonlinear Group, Dept. of Physics, University of Toronto; image source; larger image