Different Types of Solfege
Depending on where you study music, you will come across different solfege systems. Each system has its advantages and disadvantages, as further explained.
Moveable "Do": This type of solfege assigns the note "do" to the 1st scale degree of whatever key you are playing in. In other words, if your piece is in A major, do = A. If your piece is in f# minor, do = f#. This system is great for theoretical purposes, as each syllable directly represents each scale degree. Identifying leading tones, accidentals and modulations are made very easy with this style of solfege. It requires perhaps a little more thought when sight singing more complicated music that contain chromatic notes or many modulations.
Fixed "Do": In this system, the note 'C' is always Do, 'D' always Re and so on. Because of this, communicating a melodic idea becomes very specific; the solfege is a language in its own way. For example, you could say "Do-Mi-Sol" to a person who knows no English and they will know that those notes are C-E-G. The disadvantage is that scale degrees are not the same for each key. Example: Do-Sol are the 1st and 5th scale degrees, respectively, of the key C major. Do-Sol becomes the 3rd and 7th scale degrees, respectively of the key of A major. This inconsistency creates more difficulty when trying to analyze a piece theoretically.
Inflections: Another aspect of solfege is the syllable inflections. The syllable is altered if it is an accidental. Example: If you sharp Do, it becomes "Di"; if you sharp Fa, it becomes "Fi". A flattened La becomes "Le" (pronounced "lay"); a flattened Sol becomes "Se" ("say"). Not all systems use this, but it is helpful for identifying notes that don't belong to the scale diatonically.
Each system is very useful in many ways. It is definitely encouraged that you experiment with either style, but from a teaching and learning standpoint, I would suggest starting with moveable Do. Students need to work both on singing in tune as well as their theory proficiency; moveable Do helps both simultaneously in a profound way!