I don't think learning pieces because others consider them fundamental is a good way of developing musicality. Others will probably disagree, but I would tell you to learn and dissect music that you think is good instead. Your musicality should be an extension of yourself and you taste in music.
So my tip would be to listen to a bunch of songs in those genres and pick out something you really enjoy.
I agree with Luff for the most part, but everyone should at least be familiar with the basic 12 bar blues and rhythm changes, and have enuff theory under their belt to transpose them to any key.
Check out Todd Johnson's "Playin' Through the Real Book" download series. Todd's got a forum here on TB where hes talks about it. Cool examples of bass lines, solos, and chord playing on some of the most common jazz tunes.
