proxy::dist() in R in case you want to quickly try some of them.

Frechet distance is interesting, too:

"Imagine a dog walking along one curve and the dog's owner walking along the other curve, connected by a leash. Both walk continuously along their respective curve from the prescribed start point to the prescribed end point of the curve. Both may vary their speed, and even stop, at arbitrary positions and for arbitrarily long. However, neither can backtrack. The Fréchet distance between the two curves is the length of the shortest leash (not the shortest leash that is sufficient for all walks, but the shortest leash of all the leashes) that is sufficient for traversing both curves in this manner." (from Wikipedia)

I remember trying it out as a way of quantifying how two related stocks' price movements can diverge (just as an example application).