How to copy symbolic links?

cp --preserve=links

From the man page:

   --preserve[=ATTR_LIST]
          preserve  the   specified   attributes   (default:   mode,owner-
          ship,timestamps),  if  possible  additional attributes: context,
          links, xattr, all

Personally, I use cp -av for most of my heavy copying. That way, I can preserve everything - even recursively - and see the output. Of course, that is just personal preference.

As to why your other options did not do what you expected, -s makes a link instead of copying and -L follows the links in the source to find the file to copy instead of copying the links themselves.

Just as the man page says, use -P. This setting says:

-P, --no-dereference
       never follow symbolic links in SOURCE

If the links contain relative paths, then, copying the link will not adjust the relative path. Use readlink, with the switch -f to follow recursively, in order to get the absolute path of the link. For example:

ln -s $(readlink -f old/dir/oldlink) new/dir/newlink

If preserving the relative paths is what you want, than the option -P of cp, as said by Ignacio Vazquez-Abrams, is what you need.