The nucleus is the solid portion that astrophysicists say is the kilometric-sized object. There are only a few comets which have nuclei larger than tens of kilometers. The amount of comets sharply increase as they get smaller but these are harder to see. The only nuclei ever observed was of Halley's comet. This is when a probe got less than a 100 miles of the surface.
Cometary orbits come in two classes.
These have periods less than 150 years, and have orbits almost completely within the planetary system. These seem to have been captured in the solar system by the major planets. The second class has nearly parabolic orbits with periods of themselves at over 150 years. These are at random and come from any direction. Many of these have only been seen once.