I have spent a bit of time analyzing my favorites, and I usually go about it by asking :
what are the chords for this phrase, and how is the bassline supporting them?
which notes are chord tones, which are non- chord tones?
Is a scale or mode I recognize being used?
Which non- chord tones are diatonic scale tones, and which are chromatic?
Are the chord tones /non chord tones on strong beats or weak beats?
I try to seperate rhymic analysis from harmonic as much as possible...
I a rhythm I recognize being used? Is it a Cuban tumbao? a 12/8 blues shuffle?
( I recommend
Paul Westwood's Bass Bible for a useful catalogue)
How does the bass rhythm relate to the snare or hi-hat?
Is there an anticipated downbeat?
Can I notate the rhythm? (sometimes

)
in general I'm always trying to see how the bass line is supporting the harmony and thus the song....
Also, most of my songbooks are peppered with little " I-vi-ii-V7" chordal analysis so that I can understand the sing harmony in a functional way.