So if you've been following along you are probably way ahead of me on this one. If variety and contrast are part of what makes design work...wouldn't it be a good idea to use a variety of types of shapes?
Obviously a character who is all one type of shape - like made up of all circles, and nothing else - is going to be pretty boring. No contrast, no variety, a real wasted opportunity.
Take a look at this drawing by Milt Kahl (is everyone sick of this drawing yet?). I created a diagram to show how many different kinds of shapes there are in his construction.
So it creates more visual interest to have a variety of shapes at work in any drawing.
The hard part about this is making it all feel unified. When all of the shapes are different, a drawing may start to look like a bunch of mismatched parts. The key is to make all the shapes pass into one another and lock together in a natural way. All I can say about Milt's Rhino is to take a look at the outside line around the figure. The outside contour has a wonderful rhythm - it passes over each shape and flows from one shape to another. Sometimes the shapes flow from one to the next and sometimes they overlap a little and sometimes they tuck into each other - in any case all of the shapes feel connected and unified.
A big part of making any drawing hang together is to give it an overall rhythm - the rhino has a great rhythm from the top of his head to his toes. Giving your drawings an overall unifying rhythm and making all the shapes fit into that rhythm helps make it feel like all of the "parts" are flowing along the same line. So rhythm helps make them feel unified (I know that probably sounds nebulous, but I will attempt to talk some more about "rhythm" soon).
In the beginning Disney design seemed to be based on all bean (or pear, if you like) and circle shapes (think the Seven Dwarves). But over time their designs got more variety of shape and became more sophisticated and interesting (in my opinion anyway).
I think Ariel is a more interesting design than your usual Disney princess because she has round shapes for hair, an angular face, torso and arms, rounded hips and triangular fins down below - a nice assortment of different shapes.
Don't ask me why I include Wart as a bird as an example, other than I just did some screengrabs for a friend who needed reference. He's nice and appealing and an interesting mix of kinds of shapes. I particularly like how his legs and feet aren't really "shapes" per se - they're just lines. Very sophisticated for a Disney design. And I like how that puts all his bulk up high on thin legs - his bulk "floats" up in the air like a bird in flight. And his beak is just simple triangles - very bold.
Just to prove that I use stuff other than Disney as examples, some Provensen stuff with an awesome variety of shape. But mostly what I own is Disney stuff so that's mostly what you get here.
Please remember that the point of all these posts is that these design elements will improve any drawing. These concepts are not just for character designs and visual development - these are the elements that make a good drawing, period.