How do you flatten 18g silver wire?

There are some variables here - is it dead-soft, half-hard or hard? soft or half hard would be fairly easy to flatten with a planishing hammer or by running it thru a rolling mill. hard wire should probably be torch annealed first.

Another Answer:
If you don't happen to have a planishing hammer, rolling mill (!), or torch, which is totally OK if you don't, you can use other items in a pinch. Two plain old hammers work just fine. First find a level, steady and sturdy work surface. I recommend the ground, on concrete or pavement if you can kneel comfortably. Take the first hammer and lay it down on its side. Then lay your wire on the flat part. Hit the wire with short, firm strokes; its always best to start light. If you don't want to nick the wire, tightly wrap one or both hammers with tight woven cotton fabric- a cotton bandanna or old pillowcase. Hey, it won't be perfect but it does the job. If you plan to do more wireworking in the future, then maybe I'd recommend getting a small jeweler's anvil along with a chasing hammer or other hammers of choice. Unless you really want to get a rolling mill. Hope this info helps!