There are usually two things that contribute to songs not appearing under Artist on the iPod, both due to information either in or missing from the Get Info>Info tab against the song in iTunes. 1) the song is marked as part of a compilation or 2) the song has no album name associated with it.
If the tracks are ticked in the Get Info>Info tab as part of a compilation, they will not appear under Artist on the iPod. Highlight a song in iTunes that you know is not appearing correctly on your iPod. Right click on it and choose Get Info and click the Info tab (alternatively choose Get Info from the File menu), if "Part of a Compilation" is marked untick it and afterwards update your iPod, you should find it now appears as normal. Compilations are stored in a separate folder in iTunes so the songs don't appear inside the normal Artist>Album folder.
Also if tracks are missing an Album name in the Info tab, this information needs to be filled in. The no album one is a bit quirky as the songs will appear under certain circumstances but the simplest remedy is just add None or Miscellaneous to all the blanks in place of a title. See this link for more detail:
I don't see some of my songs when browsing by artist on my iPod
