In fixed Do solfege, the singer always sings Do as some form of C, whether C, C#, or Cb. Namely, Do has been “fixed” in place and no longer moves around to other notes.
Only the syllables Do, Re, Mi, Fa, Sol, La, and Ti get used. Fixed Do solfege lacks special syllables for sharps and flats. This means that you lose all the advantages of singing chromatic solfege with shape notes.
The big advantage to fixed Do solfege centers on its ability to deal with music that contains a lot of accidentals. It can be especially effective when applied to music that lacks a definite key center.
This means it remains a good option for advanced singers. However, younger students find it difficult to work with.
I had a mixed experience with fixed Do solfege at UMass Lowell. Because one syllable could represent three different sounds, I sometimes felt that I sang the notes despite the syllables. That the syllables held me back more than they helped.