I always use the lower case /y/ but it looks like phonologists do not agree about how to represent this vowel.

See the articles discussing [ʏ] on French Wikipedia versus English Wikipedia.

The Canadian French pronunciation might also play a role here:

The lax allophone of a high vowel may also appear in open syllables by assimilation to a lax vowel in a following syllable: musique ‘music’ can be either [myzɪk] or [mʏzɪk].

In France, using [ʏ] would betray an English accent and using [u] a Spanish/Italian/Portuguese one one. Germans do pronounce the [y] correctly because it is part of their phonetics (ü / ue / y ). On the other hand, the French vowel "u" is foreign to English ears so what is used is the closest familiar vowel which happen to be [ʏ].

As I learned it in my linguistics undergrad, the usage is systematic. Your best bet to cover most cases is to pronounce it [y] in open syllables and [ʏ] in closed syllables.

In European French, this sound is always [y] — at least nominally.

However, in every dialect of French, there is a tense/lax realization of certain vowels, where the tense variant appears in an open syllable and the lax version in a closed syllable.

This rule can’t be applied universally since some high vowels lack lax variants in European French: /i/, /u/, and /y/. But in Québec, these vowels do have lax variants: [ɪ], [ʊ], and [ʏ] respectively.

The rule is the same for these pairs as for others:

ici [i] vs. icitte [ɪ] *

roue [u] vs. route [ʊ]

du [y] vs. chute [ʏ]

As the Wikipedia article jlliagre shared says, a lax vowel in a closed syllable can also harmonize a nearby tense vowel, e.g. [ɪ] in musique harmonizing [y] to [ʏ]. (This effect is less predictable.)

_{* An undergrad paper I wrote hypothesizes that innovative forms like icitte and frette, whose origins are hard to identify, could have co-evolved with these lax vowels, as if to produce more occasions to use them and thereby further differentiate Québécois French for sociolinguistic reasons. Take that for what you will. 🙂}