What kind of cookies can be shaped? I was wondering what to look for in a cookie recipe if I want to cut it out into shapes using a cookie cutter.

Some cookies tend to spread out much more than others and so would not retain any semblance of the original shape. I did look at this question, but that only mentions that cookies with more fat in them will result in a more spread out cookie. What kinds of cookie dough does it makes sense to try and shape and can any cookie recipe be modified to make it more structured?
Cookies for shaping are usually rolled cookies.

Common categories would include:


Sugar cookies (as Mien mentioned)
Gingerbread
Linzer cookie dough


Look for recipes where you use a rolling pin.  These are already formulated for minimal spread.

Adapting other cookie recipes for use with cookie cutters will be trickier, and probably not worth the effort, in my opinion.