Does the water level in that stream fluctuate much? It looks like the bottom of the icicles may have formed in contact with the water surface which has since receeded.

From the look of the image, the little 'suction cups' appear to form at an approximately uniform distance from the surface of the water. This makes some sense, if my understanding is correct.Icicles tend to form ripples instead of just flowing straight down. Those ripples are formed when the water flowing down is more or less likely to freeze (it's called Laplace Instability, and it's a long story). Ultimately, the little ripples form as water alternately releases heat more quickly (it freezes quickly, building the ripple) or more slowly (it doesn't freeze so fast, and flows along, dripping off the end). Now, we add one more thing: the air close to the surface of water, particularly running/churning water, will tend to have more moisure in it and can soak up the heat more readily.As the water flows down the icicle, creating ripples, until it gets to where the moist air near the water. The ripple starts to build normally, but since the moist air can take the heat more quickly, it keeps building outward. As it gets farther from the center, it freezes a bit more slowly (the center of the icicle helps the freezing), and so gets a little farther down before freezing and builds a suction-cup.Basically, the suction-cup is a big ripple. I don't know how it would shape as it gets nearer the water, but it's possible that it will cool so efficiently that the icicle will get thicker where it meets the suction-cup part.I found a couple of sites that might help.This one makes a quick description:http://www.aip.org/pnu/2002/split/613-2.htmlThis one is a longer presentation, but has a better explanation of Laplace Instability and some examples:http://www.uwec.edu/physics/lockhart/Patterns%20in%20icicle%20formation.pdf

surely the bases are all at the same level and shaped like suction cups because they formed against a surface of ice when the stream itself was frozen over?

Here are some ice bells that are similar http://epod.usra.edu/archive/epodviewer.php3?oid=408384

The branches were being moved in circular patterns by the breeze and the water droplets were swayed away from center by that motion. Centripetal force frozen in time.