The range of the basic scanner is 500LY so it will find almost all asteroid rings and close-in planets. You may miss half the system or more, however, if you don't go exploring further out. It will also usually not reveal stars other than the main star in the system, as with the exception of close binary systems the other stars orbit or co-orbit at a considerable distance - 10s to 100s of LY out.
The intermediate scanner has a range of 1000 LY, and the advanced scanner finds everything. This can be an astonishing difference, sometimes, as the main star is not always the one with the largest number of satellites. Eg I surveyed one system near Ethgreze as thoroughly as I could with a basic scanner. I found about a dozen planets and asteroid clusters, and I knew there were two distant stars but they didn't come up on the nav panel. When I went back later with an advanced scanner it discovered the location of the stars plus an extra 58 celestial bodies!