Is there a curvilinear plug-in or script?
What do you mean? Can you post a sketch of what the plug-in is supposed to do?
I have some handwritten characters. When I scan them and turn them into pictures and then turn them into outlines, there are many jagged edges, all of which are points and line segments. Is there a plug-in that can make the contours smooth or curvilinear, making the characters look more refined rather than rough
That depends largely on your vectorisation settings. Feel free to experiment with different algorithms, like the ones in the Trace Filter.
But a scanned result can at best just be a good guess and will usually require manual intervention.