Do they have to be removable on a regular basis? If not, mylar (Melinex) straps would be an easy and economical solution...
Clear polyester film straps can work either way. For permanent attachment use two narrow (say, 1/8" wide) straps; one where the bowl of the spoon meets the stem, and one at the end of the stem. By this method you can arrange the spoons at all angles.
For removable attachment of the spoons mounted stem-down, you could use a single strap at least 1/4" wide on each spoon, placed just below the highest point of the spoon stem's curvature, and install it loosely enough that the spoon could be lifted out and the put back. If the spoons are similar in size and shape, the rows of spoons could be positioned easily, but if the spoons are different shapes and sizes, then it would take more time to locate and install the straps of various lengths.
maybe use loops of elastic cord
Be careful with this. Most of the times I have used elastic cord in the past, the elasticity deteriorated over time and the cords had to be replaced within a few years. The round-profile cords are quite visible, too.
If you decide to use any sort of monofilament mounts, make sure the material is polyester and not nylon, because the nylon exposed to light would deteriorate and weaken. Using 97% or better UV filtering glazing would help, but that would remove only the radiation from 280 to 380 nm. The visible light spectrum from 400 to 700 nm is damaging, too, and that wavelength range can not be blocked.