I was taught fixed do as an undergraduate. Later on, when I had to tech ear training, I adopted the movable do system. I think either works - both have their pros and cons. I think the most important thing is that the work be done consistently, over time. No magic bullets, here. In the end, all roads lead to Rome, so to speak. I know fixed do-ers who are brilliant, I know moveble do-ers, and I know people who never adopted a system, yet their ears are very good. Practice, listen, practice, listen; repeat many times...
Sight singing is taught in University or Music Conservatories as part of the Ear Training curriculum. It is where you sing a melody having only been given the tonic or the first pitch using some type of system of syllables.
In some parts of Europe, for example, pitches are referred to by their names in solfeg (do, re, mi, fa, sol, la ti, do) instead of A, B, or C. When they sight sing, they often use those syllables.
DMA Student and Teaching Asst in Music Theory/ Composition at the University of Miami Personal Website
When I was in conservatory they tried some moveable do on us. I did not succeed too well because at that time I already had a trained perfect pitch in the middle range. Letters, intervals and numbers were ok for me.
As a choirboy I was taught the movable doh system as well as staff notation. I could read equally well from either system. And at that time many publishers issued partsongs and church music in staff notation with tonic sol-fa names below each note.
When I became a school music teacher I also taught both systems. Tonic sol-fa first (movable doh), then how to apply it mentally to staff-notated music. I had excellent and consistent results and turned out a lot of efficient singers and sight-readers.
Incidentally I am a firm believer in the "lah" minor, not the "doh" minor. In other words the harmonic minor scale is l. t. d r m f se l This relates perfectly to the way we notate the minor key in staff notation.
I was taught ear training in university which was very effective for me. I was also taught solfege but never use it. I found learning a 'system' to learn a 'system' (musical notation) really cumbersome, tiring and pointless. These days if I have reach to get a note, or use my inner ear--I just intervals to help out.
One point in favor of movable do is that once the tonic has been determined, any interval is easy to sing/hear. Where this comes in handy is using clefs other than the one your primary instrument uses, and especially for the movable clefs. Still, many fine musicians learned the "European" fixed do method, and they turned out just fine.
Related story: a friend of mine taught theory and ear training at a performing arts high school, where one of his colleagues was from Russia. She had been firmly schooled in tthe fixed system, and had little tolerance for the movable approach. When asked, she said (insert Russian accent here), "Well, if you use movable do, then even when you modulate, you are still in the key of do - you are still in the same key, all the time. Just like Yanni."