I play in two bands with vocalists:
- a Big Band, which usually (but not always) alternates between 'vocal sections' and 'instrumental special sections'; the golden rule is that when then vocalist is singing, the band plays 'softly'. Kind of a supporting 'blanket' for the vocalist who's show it is. On the 'specials', the band can let rip until the next vocal section
- A pop/rock band: some songs (Blues, Funk) lend themselves are great for sax fills and/or solo's, others not so much. FWIW I try to listen to song beforehand (YouTube: original and Live) and discuss with the band whether, when and how my sax can add anything (or not). In our band, it's often the keyboard player who adds short fills between vocal lines.
In my limited (2 month experience) it depends a lot on:
a) do you have a keyboard player? if so, his/her 'sound' (piano <-> hammond) can either complement or compete with the 'sax sound'
b) does a sax really add anything to the song/band?
c) the duration of the 'gap' between vocal lines (is there enough time to play a 3-note fill?)
d) is the 'gap' already filled? (keyboard/guitar?) and can you 'renegotiate'?
e) the 'dynamics' of the song; I don't necessarily start playing in the 1st verse/chorus but a sax in the 2nd verse/chorus can give a bit more 'body' to the development; Similarly a song sometimes reaches a crescendo (in volume) or a change of tempo/key where the addition of a sax gives an additional impulse
f) after xxx guitar solo's, a sax solo can add something new and refreshing
g) don't forget that a slow sax intro/outro can be effective too!
Just 3 of examples from our repertoire:
- extreme example:
great balls of fire (piano-centric, fast rhythm/lyrics) sax doesn't match tune and no room to play anyway!
- Unchain my Heart (Joe Cocker) sax fills + solo
- Pearl's a singer (Elkie Brooks): I chime in at the tempo change (2:50) but the sax player plays 'supporting notes' during most of the song
View: https://youtu.be/0R9c5fFVtBI?si=uTvj97m0DxF3lBCH
View: https://www.youtube.com/watch?v=v2yYLdigGbo