Theory # 1:
You know how if you take a balloon and rub it against the sofa or your head and get it all staticy and then turn on the faucet just a bit so there’s a thin stream of water coming out and then hold the balloon near the stream of water the balloon will actually push the stream of water over a little? Well, maybe there’s a bit of static electricity inside your frizzer freezer on the bottom that’s pushing through the tray, and making the little icicle on top.
Theory #2:
Right when the water is freezing, (it will first freeze on the top, and on the edges, moving toward the center,) something startles the tray. Maybe someone jumps on the floor above, or maybe you just knock against the freezer as you’re passing by, which causes a small drop of water to shoot up through the part of the top layer of ice that hasn’t yet frozen, and, since it’s such a small amount of water, and because it’s already so cold, it freezes before it can make it’s way back down below the surface.
Theory # 3:
What happens when you fill up a pot of water from the tap? Aren’t there always a few tiny air bubbles that get stuck to the surface of the pot under water? Those form in the ice trays when you fill them up as well. As the tray it sitting in the freezer, those bubbles gradually start to come together, and form at the surface. When the water freezes, the air bubble in the center forms the icicle.
Theory # 4:
When the water is in the tray, the water level is higher on the sides of each ice thingy, and lower at the meniscus, where the spike later forms. But as the water freezes, what happens to the water molecules? They become rigid, and push against the sides of the ice tray thingies, and expand. So what happens to the water level? The middle, due to the pressure on the sides of the thingy, actually push the center of the water level up, so that at the very center of the cube a spike forms.