Any shape you can imagine is possible as long as I can manipulate it in my saw.
Shaped puzzles can be produced with no image for an extra challenge. Not only will there be no image clues to direct the assembly, you also won't know if the pieces are right-side up!
Pieces may be cut in a grid pattern like this...
or with more of a random shape and orientation...
I think that both patterns are attractive. The random shape makes more sense if there are a lot of silhouettes or color border cuts involved.
The number of pieces per square inch is known as the density. A density between 1.0 and 2.0 seems most natural to me for the random shape, while densities approaching 3.0 are achievable with the grid-shaped pieces. You may multiply the density by the puzzle area to estimate the number of pieces.
Cutting pieces along color borders makes the puzzle more challenging to assemble. There are two styles of color border cutting. The most challenging to assemble but least interlocked option has the cut completely on the color border. The highly interlocked option with slightly easier assembly has lobes protruding from the color border as shown.
Specially-shaped pieces may be cut from a puzzle. The shapes may be used to personalize the puzzle, as with letters and numbers forming names or dates. The shapes may have themes relating to the puzzle image, such as a Christmas stocking shape in a wintry scene. Clever selection and placement of the silhouette pieces can add a lot of character to a puzzle.
When larger silhouettes are used, they may be cut into several pieces.
Color border cuts and multi-piece silhouettes taken to their extreme can create a sub-puzzle. A sub-puzzle is a puzzle within a puzzle that can be displayed on its own. I really like these because you can choose to store the sub-puzzle pieces seperately so you can assemble it by itself if you choose.
A puzzle may be cut into very large pieces and then each piece subdivided into smaller pieces. I particularly like this technique because it increases the assembly difficulty by creating many false edge pieces and reducing the overall interlocking of the puzzle. The finished puzzle is also visually striking.
The time required to solve a puzzle increases exponentially with the number of pieces, so a 200 piece puzzle will take more than twice as long as a 100 piece puzzle.
If you enjoy the challenge of a large puzzle but still want to retain the opportunity of assembling a small puzzle, then a multi-puzzle is for you. A multi-puzzle is simply multiple small puzzles having similar color schemes. The small puzzles could all be from the same artist or you could even buy multiple puzzles all using the same image. Since the puzzles are hand-cut and unique, each piece will fit only in the puzzle from which it was cut.
The romance novel illustration series on my available images page are good subjects for a multi-puzzle. Pick out your favorites and I will cut them all with the same style to create a real challenge for you.