Which languages other than some Chinese languages have apical vowels? The "apical vowels" are the i in zi, ci, si (in IPA: z̩ (also seen as ɿ)) and ʐ̩ (also seen as ʅ). They are basically buzzed sounds after certain consonants.
The Mandarin apical vowels are examples of syllabic fricatives — that is, fricative sounds that form the nucleus of a syllable, as a vowel generally would. (Sometimes syllabic fricatives are also called "fricative vowels," though that's rarer.)
Here's a discussion from one of Ladefoged's books on "fricative vowels" a.k.a. syllabic fricatives in other parts of the world. He mentions examples from Czech and Bantu. Berber is also often described as having syllabic fricatives. Blackfoot has recently been claimed to have syllabic /s/.
(It's maybe worth mentioning too that you'll hear syllabic fricatives in English, if only as a fast speech phenomenon. So for instance, in my dialect, the word horses is [hɔɹsɨz] if I'm speaking carefully and [hɔɹsz̩], with a syllabic [z̩], in fast speech. If I'm in a hurry, the last syllable of horses sounds an awful lot like Pinyin si.)
According to Wikipedia, there are other Sino-Tibetan languages with apical vowels. But I don't think they're very common at all: out of the major Chinese varieties, only Mandarin has it.
（例）ピㇲトゥ pɿtu（人） カビㇲ kabɿ（紙）
The name of the language itself contains an apical vowel: [mjaːkufutsz̩].
I had the opportunity to talk about this issue with a Chinese linguist who is teaching a class I'm auditing at Fudan University.
Chinese-linguists have traditionally used the term "apical vowel" (and the otherwise unseen characters ɿ and ʅ) to describe the nucleus of the "zhi chi shi ri" and "zi ci si" syllables. The textbook we used was no exception.
As others have pointed out, these are often analyzed (or at least written as) syllabic fricatives by the linguistics community at large. I wasn't sure if my professor knew this, so I brought it up with him after class.
He was actually quite insistent that there is no frication in nuclei of these syllables (and was able to pronounce the syllables in a way consistent with that hypothesis). We eventually agreed that there was a fair amount of variation among Chinese people as to the realization of these phonemes.
So, if and when they're produced without frication, what are they? The retroflex one can be analyzed as a syllabic-retroflex-approximant. The apical one is harder, as both the front and the middle of the tongue are raised quite close to the roof of the mouth. A bit like a [ ɨ ] with the front of the tongue raised slightly.
It is argued in this paper that the appropriate analysis of Modern Standard Mandarin "apical vowels" is as syllabic approximants:
- the vowel in pinyin zhi chi shi ri (traditionally ʅ) is argued to be ɻ̩ (ɻ with syllabic diacritic)
- the vowel in pinyin zi ci si (traditionally ʅ) is argued to be be ɹ̪̩ (ɹ with dental and syllabic diacritics)