At Transformed Puppet, puppeteer and artist Ronald Binion has an excellent page about issues to do with making the interior of a foam puppet mouth both comfortable and effective in manipulation. He illustrates the kind of hand grips that he prefers.

As he notes, the interior is just as important as the exterior aesthetics in terms of using the puppet expressively; and as there is a propensity for puppeteers to suffer from repetitive strain injuries in the long run, it is also an occupational health issue that puppet makers have to keep in mind. Binion is also right in suggesting that there are many solutions to the problem. One of the joys of puppet making is that it is about problem solving, and individual approaches are valuable and part of the art. I particularly like the look of the thumb tube that is ‘connected to the mouth plate at only one point, allowing the thumb tube to act as a lever, and not force the thumb to conform to the angle of the mouth plate’. It’s also vital that the hinge of the mouth fits snugly into the hinge of your hand, between your thumb and fingers.

Take a tour around the Transformed Puppet (updated link 2015); there are some interesting puppet pictures and productions there. (Via PuppetVision Blog)