Very odd problem with songs not showing up

Two things, both are well-known issues:

1- Make sure that all songs have an album name. If you don't know it put "Unknown" or "N/A".

2- Check to see if the songs are tagged as "Part of a compilation". If they truly are part of a compilation (a mix of various artists) then you might leave them tagged that way. If they are on a "best of" album, you might consider not tagging them as part of a compilation -- I prefer to designate compilations as a mix of artists. If you change your iPod setting for compilations to OFF, then all artists and their songs will show up under the Artist menu. If you turn your iPod's compilation setting ON, then artists who only exist on one compilation album will be removed from your Artist menu. That is Apple's way of "cleaning up" your list of artists by removing the ones with only one song in your library.

Sort your library by album, group-select the ones with no album name, select "Get Info" and enter something for the whole lot of them.