Nina Paley has some really interesting vector animation techniques and one that caught my eye is a kind of waving tapered line. I've attached an example from her blog. She uses Flash for this but I'm wondering if it can be done in AE.
I tried the Ripple effect (with a non-zero wave speed) and got some interesting motion but it doesn't seem the same. I have a feeling it's something simple that produces such a mesmerizing effect but I can't figure it out.
Of course, right after posting my question I found the solution on a second search of the forums. The Wave Warp effect does the trick and the "wave type" makes for some interesting variations. Here's my quick-and-dirty version, which came out more ominously but shows the effect.