I'm taking an ear training class at a community college and when we sing we sing numbers representing scale degrees. I think this is preferable to Solfege for jazz because we already think scale degrees.
Here's part of a page from our book. We are one on one tested on this stuff and the teacher will pick a page from the book and ask us to sing it or she'll write her own melodies and ask us to sing it. (It's sight singing, no instruments are used, no reference pitches, she doesn't care about the key just that you get the relative pitches correct). Pretty tough tests.
So take example 7 which is in Ab. We'd sing "3 4 5 6 7 1 3 2 5 4" etc. at measure 9 we'd sing "5 5 6 7 1" and knowing in your mind that the second 5 is a sharp 5 so you'd sing two different pitches for the 5, in your mind you're thinking 5 #5, while singing the words "5, 5" i.e. same words but different pitches.
And the next step or level of development is to just think the scale degree numbers in your head while singing a sylable like "la", for some reason this is more difficult for most, including me.