Harmonics occur at the whole-number ratios of the length of the string, e.g. 1/2 the length of the string, 1/3 the length of the string, 2/3 the length of the string, 1/4, 3/4, etc. The lower the numbers involved, the easier the harmonic will be to play. So for example, theoretically there's a harmonic at 6/19 the length of the string, but it will be much harder to play than the one nearby at 1/3 the length of the string.
To determine the pitch of the harmonic, just divide the open string's frequency by the harmonic's string-length ratio. For example: the 7th fret is 1/3 of the length of the string (from the nut), so if the open string's frequency is X, then the frequency of the harmonic at the 7th fret is X/(1/3) = 3 X, which is an octave-and-perfect-fifth above X. Two harmonics who's string-length ratios have the same denominator (when in lowest terms) have the same pitch, so for example the harmonic at 2/3 the length of the string will also be an octave-and-perfect-fifth above X.
The most common harmonics occur at:
- 12th fret (ratio 1/2): an octave above the open string
- 7th fret (ratio 1/3): an octave plus a fifth above the open string
- 5th fret (ratio 1/4): two octaves above the open string
- 4th fret (ratio 1/5): two octaves plus a major third above the open string
- 9th fret (ratio 2/5; equivalent to 1/5): same as at the 4th fret, since the ratios have the same denominator.
Those are all I can think of right now; feel free to mention other common harmonics in the comments and I'll add them.
All of this works for pinch and touch harmonics as well, since when you fret a string, you're basically making it into an open string with a shorter length. The harmonics will all be in the same places relative to the length of the fretted string. In practical terms, to determine the fret at which a fretted string's harmonics occur, all you have to do is add the fret number to the harmonic's fret number. For example, if you fret a string at the 4th fret, then the "12th fret harmonic" will now occur at the 4 + 12 = 16th fret, so there will be a harmonic at the 16th fret with a pitch one octave higher than the fretted note.