This is sooo weird. As is my habit, I filled my ice trays with cold tap water and let them settle on the counter for 15 minutes or so. Then I put them in the freezer. A few hours later I opened the freezer and found that the ice had these nail-like projections. The longest was maybe 3 inches long. They weren't touching anything. What the heck happened? Notice the baby dude in the first picture? WTF?
They're often called ice spikes. Ten years ago these things were pretty mysterious, but now Wikipedia even has a video of them growing out of ice cube trays in a freezer.
Basically, the ice surface freezes first, which slightly pressurizes the water underneath. That water breaks through the crust, and continues to flow, and freeze to equalize pressure within the cooling cube. Temperature conditions have to be just right for it to happen, so you don't see them all the time.