Why can't you just make a cut from the edge of the paper to the thing you want to cut out, then cut it out? Are you trying to end up with a stencil rather than with a silhouette of the item itself?
An x-acto type knife blade would be the tool for the job (if you didn't use a machine like a Cricut outfitted with a die of the exact shape you want to cut out). But to cut "cleanly" with an Xacto or other craft/etc knife, you'd need:
... a sharp blade (perhaps a new blade)
... one of the pointed blades that fits in your knife
... a proper cutting surface (e.g., a "self-healing mat" or perhaps even a sheet of glass)
... also moving the paper (or the whole surface) as cutting, or with individual passes
click on VIDEOS near top of page)
You could use the tip of a single-edge razor blade but that's much more difficult to maneuver well since the whole blade is short and wide, and especially for anything but straight lines....a long handle works much better.