Most asteroids are too faint for amateur instruments to detect, Still, there are several hundred within reach of a 3 inch (75 mm) telescope. For most observers, the question is: where do I look?
Astronomical almanacs publish positions, called ephemerides, for the brightest asteroids, and more can be found in computer software, astronomy magazines, and annual astronomical hand-books.
When you have positions for an asteroid, plot them on a star chart and observe the field. Start with one of the bnghter asteroids—these just reach naked-eye visibility. When you locate the right area, compare your sky chart with the star field using your lowest-power eyepiece. If you are lucky, the asteroid will be easy to identify as the "star" not plotted on the chart.
If the asteroid is not obvious, you will have to identify the most likely area for it to be in and plot all the star-like objects you see, going down one magnitude fainter than that predicted for the asteroid. Wait several hours, or come back the next night, and re-observe the field. Your asteroid is the star that moved.
From time to time, an asteroid will pass in front of a star, briefly blocking its light. Such events, called occultations, are extremely valuable in determining an asteroid's physical size and shape. Often it is the only way to gain this information. Amateurs join professionals in expeditions to locations where they hope to see the event. As it is difficult to know precisely where the best view of the event will be visible, the more observers there are, the better. Anyone interested in helping with this kind of observing should contact their national amateur organization