The first thing I thought of was the Eames Hang-it-all coat rack. It is bright and whimsical with long enough posts for puppets. The brightly colored balls at the end of each post would also add a cute touch intermixed with the puppets-- then you could hang dress up clothes or capes and hats on it later when/if the puppet look is no longer desired.
Here's a photo of it from one website:

http://www.allmodern.com/Herman-Miller-%AE-HIA-HML1164.html

What about an over the door shoe rack...
like this?

We had only a few puppets. We displayed them on the bedposts.

I bought a puppet stand at a local teacher supply store. It's been great. We put all of our finger puppets (which we have more of) on a wooden stand meant for rolls of thread. It's worked perfectly and looks great too!

LOVE.

Great ideas...thanks.

I just picked up the computer to order a puppet stand, as today, the last day of Christmas, my kids received their last presents -- 12 hand puppets.

I'm probably going to buy these:

http://www.ivgstores.com/IVG2/Y/ProductID-51866-.htm

I know there are some for finger puppets too.

You could make a puppet stand by buying a piece of wood and a dowel. I'm sure Michaels or some craft store sells round or square finished pieces of wood, drill a hole, glue dowel - voila!

Well, this idea does not mount to a wall, but looks great on a shelf. IKEA sells the IDBYN mug, that is a large plastic tumbler. It is great for putting upside down underneath the puppets and making them stand up! They are pretty inexpensive, I think .99 each.

For finger puppets, I bet a thread rack (meant for storage of thread spindles) from your local craft store would be perfect!

I actually had an old paper towel rack (stand-up, not wall-mounted, though) that got drafted as a puppet stand. It sits on the floor and works just great!