Angel, have you tried putting a sphere or something in the mouth to grip, rather than the straps?  I use the gasket rubber instead of straps in some of my puppets. In others I use a "sphere." 

I have not tried a sphere.  I do put a little "bump" of foam under the strap.  I actually use a piece of the foam that you use to cover pipes with.  I cut it about 3 inches long and then cut a slice out of it and fasten it to the mouth plate then put the strap over that.

I'm interested in the sphere idea.  Can you tell me what you use and how you use it?

Correction, I don't know if it's 3 inches, I just eyeball it! LOL

I make a pocket on the bottom and on the top I either wrap an elastic strap around a piece of foam sized to my hand and glue it, or build a similar pocket on the top if I have the room in the head.

I know some people use the insulation foam as well and make finger holes on top and bottom. What does everyone use for the mouthplate itself, say for something like Elmo or Ernie where the mouth is stiffer?

I have tried the finger tube method shown on "The Foam Book" DVD.  My puppeteers didn't like it and the tubes sometimes came unglued.

Animal, when you make the pocket, do you have problems with the hands sliding out as you work the puppet?

Not as of yet, up until the newest one I made all the heads so that the mouth's natural position was open, closing was the effort. This one I am trying is a new head design where the jaw will be much looser than before, almost hinged, so we will see........

Do you use foam or fabric to make the pocket?  Is it glued around the edges or sewn?
Sorry so many questions!

I don't mind questions at all, I'm just not sure I'm doing it right either....

I glue 1/2 inch foam to top and bottom of the mouthplate, I then draw an outline of my fingers and thumb with a sharpie and test it with both my left and right hand. On the bottom I then hot glue around the outline and the edges and place 1 inch foam in place. the middle becomes my pocket.

Be sure to make it slightly smaller than the mouthplate foam or you'll get a HUGE chin on your puppet! I've had to trim it down more than once because of this. I like the fit and security from the 1 inch foam on the thumb, the top is different, I cut the width of my four fingers and then glue an elastic strap under it before gluing it to the top.

I also sometimes just use the pin from the doll joint over my finger for support and grip if I used one to fasten the nose. I've never gone as far as to built an actual strap for the mouthplate because I use foam on it...I may have to try that sometime instead.

Keep in mind that is what I do for foam head puppets, not for sock puppets. So far I've just glued 1/2 foam to the top and bottom, sock puppets are usually fairly snug, but if the head is larger I run an elastic strap across the foam before glueing it...