This uses dhok's classification system:
T indicates a triangular vowel system, by far the most common.
S indicates a square vowel system.
V indicates a vertical vowel system.
C indicates a "cubic" vowel system.

This is followed by the number of vowels, and:
C indicates that the system mostly has central vowels as its "extras".
R indicates that the system mostly has front rounded vowels as its "extras".
U indicates that the system mostly has back unrounded vowels as its "extras".
F means it has extra front unrounded vowels as its "extras".
B means it has extra back rounded vowels as its "extras."
L means it has extra laxed vowels as its "extras".

This classification system does not generalize perfectly to all languages, because some languages, such as English, have diphthongs that pattern as single vowels. But it'll do.

One vowel

There are only two languages that are analyzed as having one vowel: Nuxálk and Moloko. In both analyses, the only vowel present is /a/. This will be called 1; one element cannot form a triangle, a square, or a line. It is just a point.

In Nuxálk, the phonetic vowels are [a i u], but [i u] may be analyzed as syllabic variants of /j w/, since /m n/ may also appear as syllable nuclei.

Moloko is more complicated: there are ten phonetic vowels, [a ɛ œ ɔ ə ɪ ʊ i y u], written by four vowel characters, <a e ə o>. The schwa (and all vowels higher than it, all of which are its allophones) is epenthetic and predictable, and front rounded vowels only appear next to labial consonants. The high vowels [i u] ([y] appears to pattern as a lax vowel, the labialized version of [ɪ]) are allophones of schwa adjacent to the consonants /j w/. As for the front unrounded and back rounded vowels, the front/back contrast is word-level, and is analyzed as a prosodic feature. So [a ɛ ɔ] are all analyzed as /a/, and [ɛ] rounds to [œ] when adjacent to a labial consonant, and all other vowels are analyzed as epenthetic and predictable (and therefore non-phonemic) schwa.

Word-level palatalization in Moloko also affects consonants, as can be seen in the verbs: verbs without any other suffix take the suffix -aʲ. So nəcəke [nɪtʃɪke] "I moved", becomes nəcəkva [nətsəkva] "I moved already" when the perfective suffix is added. (Presumably these are analyzed as /n-ʦkaʲ/ (the n- is a first-person marker) and /n-ʦkva/.)

Two vowels

S2

e o

S2 is commonly reconstructed for Proto-Indo-European, with /a/ arising in its descendants from sequences of /e/ and a laryngeal, and [i u] being syllabic versions of the semivowels /j w/. As in Nuxálk, this syllabic-semivowel analysis is suggested by the fact that the nasals /m n/ may be syllable nuclei; but Proto-Indo-European also allows syllabic /l r/. S2 is probably not attested.

V2

  ə
  a

V2 exists in the Arrernte language of Australia, which offloaded its rounding contrast onto its consonants: every consonant exists in an unrounded-rounded pair.

V2b

  ɨ
  a

V2b exists in some Chadic languages, like Margi.

Northwest Caucasian languages generally do not have two-vowel systems: they are commonly described as having a three-vowel system with one length contrast, /a: a ə/, the short and long low vowels in fact contrast in height: the long low vowel is low, the short 'low' vowel is low-mid, and the schwa is high. However, the long low vowel developed from absorption of pharyngeal consonants into the low-mid vowel (which did not occur in some Abkhaz dialects), so Proto-Northwest Caucasian may have had a V2 system.

TODO MANDARIN

Three vowels

T3

i     u
   
   a

T3, /a i u/, is a not-uncommon vowel system: it occurs in Quechua, Inuktitut, Classical Arabic, and many Australian languages. It usually occurs with a length contrast, but not always; it rarely takes other dimensions.

T3b

e     o
   a

This vowel system, lowers the high vowels of T3. It occurs in Cheyenne and Amuesha; it is unclear to what extent it is just a notational variant.

T3c

i
      o
  a

This vowel system is more common than T3b, occurring most notably in Piraha.

V3

ɨ
ə
a

V3 may occur in some languages in the Sepik family of New Guinea, and is developing in Marshallese, a V4 language in the process of merging its low-mid and high-mid vowels.

V3b

ə
a
a:

This variant of V3 occurs in the Northwest Caucasian languages; but remember that /a/ is low-mid and /ə/ is high. These languages all offload rounded-unrounded and front-back contrasts onto consonants, as is the norm for languages with vertical vowel systems.

V3F

i
ɛ
     a

However, some languages with vertical vowel systems do not offload contrasts onto consonants: Wichita has only ten consonants, and the vowel system /a ɛ i/, though [o:] arises as a contraction of the sequence /awa/.

Arapaho almost has the unusual V3Fb system /e o i/: its vowel system is actually /e o i u/, with no low vowels, but /i u/ occur in almost-complementary distribution, and there is only one minimal pair between the two.

Four vowels

T4

i     u
   ə
   a 

This vowel system occurs in the Taiwanese language Rukai, and also in Hongyan Qiang. The vowel system of the latter deserves closer inspection: though it only has four vowel POAs, it has twenty-four vowels.

i    u        uʴ    iˤ    uˤ   iˤʴ   uˤʴ
   ə       əʴ          əˤ        
   a       aʴ          aˤ        aˤʴ

     u:             i:ˤ   u:ˤ  i:ˤʴ  u:ˤʴ
  ə:       ə:ʴ      
  a:       a:ʴ         a:ˤ

Pharyngealization contrasts in Hongyan Qiang correspond to POA contrasts in other varieties of Qiang: Hongyan /i ə a iˤ əˤ aˤ/ correspond to Mawo Qiang /i ə a e ɤ ɑ/. In many varieties of Qiang, the functional load of the i~e and u~o contrasts is low, and the vowels are often merged; it is likely that their vowel systems contain pharyngealization contrasts which have not yet been described, and have fewer POA contrasts than has been reported.

T4F

i     u
  e
    a

Akkadian, Hittite, Malagasy (except not), Proto-Slavic (sort of).

T4Fb

i
  e   o
    a

Common in North America: Nahuatl, Navajo, etc.

S4

i   u
æ   ɑ

Seri.

V4

ɨ
ɘ
ɜ
a

Marshallese, but see above.

Five vowels

T5

i       u
  e   o
    a

Basically everything.

T5C

i   ɨ  u
  e   
    a

Lokono Arawak.

S5

i    u
   ə
e    a

Big Nambas.

S5C

i ɨ u
a   ɔ

Uto-Aztecan.

S5R

i ʉ u
ɛ   a

Cora.

Six vowels

T6C

i   u
e ə o
  a

Nepali, Armenian.

T6Cb

i   ɨ   u
  e   o
    a

Slavic, Guarani, Comanche. (Some Taiwanese?)

T6R

i y   u
e     o
    a

Souletin Basque. /y/ split from /u/ and then shifted some /i/ to /y/ in words with /y/. (I think.) However, the front rounded vowel may be either high or low, and has been described as closer to the French [ø] than the French [y]. Tsou has /ʉ/ instead of /y/.

T6Rc

i y
e ø o
    a

Wari'. (Is there a T6Rb?)

S6

i    u
e    o
æ    ɑ

Chamorro, Menominee, Persian, Forest Nenets. (Developed from T3 + length in Persian. Does Chamorro have this or is the back vowel rounded?)

S6F

i     
ø ə o
 æ a

Saisiyat, though æ ø may be variants of a o.

S6R

i   ɨ
  ø   o
ɛ   a

Hopi.

Seven vowels

All vowel systems with over seven vowels take T5 as a base. (TODO find exceptions.)

T7C

i ɨ u
e ə o
  a

Northern Welsh, Kashmiri, Romanian.

T7Cb

i y u
e ə o
  a

Albanian, Ejagham.

T7Cc

i    u
e    o
ɛ  ə
   a

Amharic. (really?)

T7L

i   u
e   o
ɛ   ɔ
  a

Vulgar Latin, Italian, Bengali, Brazilian Portuguese, Pohnpeian, many African languages including Yoruba.

T7Lb

i u
ɪ ʊ
ɛ ɔ
 a

Volow, Mwotlap, Vera'a, Nume, Olrat, all spoken in Vanuatu. Koro adds the diphthong /ɛa/; Dorig adds a long low vowel /a:/.

T7R

i y    u
e ø    o
    a

Hungarian (some dialects have an additional low front vowel, but the standard merges it into e).

T7Rc

i y  u
e    o
ɛ
   a

Occitan, after a vowel shift: ɔ > o > u > y.

S7R

i y u
e   o
æ   a

West Saxon Old English, after a merger of ø into e.

Eight vowels

Many of these are extended versions of T7L.

T8B

i y  u
e ø  o
     ɔ
   a 

Legion's dialect of French.

T8C

i   u
e   o
ɛ ə ɔ
  a

Javanese, Catalan, São Tomean Creole, and Slovene.

T8Cb

i ɯ u
e   o
ɛ ʌ
  a

Korean, though /ʌ/ has become rounded /ɔ/.

T8CR

i ʉ
e   o
ɛ ə ɔ
  a

Lo-Toga. It's not uncommon for /u/ to be less back than /o/; this is best seen as a slight phonetic variation of T8C. Lo-Toga, however, adds the diphthongs /ie iɛ ia oə oɔ/.

T8F

i y u
e ø o
æ   ɑ

Finnish, possibly some dialects of Old English. Note that Finnish has front-back vowel harmony (though /i e/ are neutral); Old English did not.

T8L

i   u
ɪ
je
e   o
  a ɔa

Tetelcingo Nahuatl. *a *e *i *o *a: *e: *i: *o: > a e ɪ o ɔa je i u

S8F

i   u 
e ø 
ɛ   ɔ 
æ   ɑ

Yapese.

C8

i u
ɪ ʊ
e o
a ɔ

Here we get our first "cubic" vowel system: a square vowel system with an additional variable, like laxness or roundedness. These mostly show up in Africa and Central Asia, and usually have vowel harmony; C8 is found in Igbo. This is the result of vowel harmony: /ɪ ʊ ɔ/ are one set of vowels, /a e i o u/ another.

C8L

i u
ɪ ʊ
ɛ ɔ
æ a

Lakon.

C8Lb

i u
e o
ɛ ɔ
æ a

Lehali.

C8R

i y ɯ u
e ø a o

Turkish. Vowel harmony.

Nine vowels

T9F

i y     u
e ø   ɤ o
æ       ɑ

Estonian. (Unlike Finnish, there is no vowel harmony, though some dialects have redeveloped it.)

T9Fb

i y  ʉ  u
e ø     o
ɛ       ɑ

Swedish.

T9L

i   u
ɪ   ʊ
e   o
ɛ   ɔ
  a

Maasai.

T9R

i ʉ
ɪ     
e ɵ ə o
    a ɔ 

Hiw. Some sources say /ɪ/ is absent.

T9Rb

i   u
ie
e   o
ɛ œ ɔ
  a

Lehalurup.

S9C

i  ɨ  u
e  ə  o
ɛ     ɔ
   a

Thai, European Portuguese.

S9R

i y  u
e ø  o
ɛ    ɔ
   a

Standard French.

S9Rb

i y ɨ ʉ u
e       o
ɛ       ɑ

Southern Sami.

Ten vowels

T10L

i   u
 ɪ ʊ
e   o
ɛ ə ɔ
  a 

Hindi, Punjabi, some African languages maybe TODO.

T10R

i y  u
e ø  o
ɛ œ  ɔ
   a 

Breton.

T10RB

i     u
ɪ ø   ʊ
ɛ œ   ɔ
    a ɒ

Lemerig.

S10C

i ɨ u
e   o
ɛ ə ɔ
a   ɑ

Khmer.

S10R

i   u
e ɘ o
ɛ ɐ ɔ
a   ɑ

Skolt Sami. This is probably a notational variation of S10C.

Eleven vowels

Past this point, classification ceases to be useful.

Danish

i y   u
e ø   o
ɛ œ ə ɔ
  a    

The full range of front rounded vowels is only preserved before nasals.

Gheg Albanian

i y   u
e   ə o
ɛ     ɔ
a   ɑ ɒ

Plus nasal /a e (o) i y u/.

Mwerlap

i  ʉ
ɪ  ɵ  ʊ
ɛ  ɞ  ɔ
ɛa ɔɞ ʊɵ
   a     

Note the diphthongs.

Sakao

i y u
e ø o
ɛ œ ɔ
a   ɑ  

TODO Vietnamese.

Twelve vowels

Selkup

i y  ɨ   u
ɪ
e ø  ɘ   o
ɛ        ɔ
æ    a

TODO should check this

Thirteen or more vowels

Dutch

i y       u
  ɪ ʏ     o
e ø   ə   ɔ
ɛ     
a       ɑ

Dutch TODO

German

i y    u
  ɪ ʏ  ʊ 
e ø ə  o
ɛ œ ɐ  ɔ
    a    

The nonlow central vowels only occur in unstressed syllables, and some dialects turn /ə ɐ/ to [ɛ ɔ].

Kensiu

i ɯ u
ie
ɪ
e̝ ɚ o̝
e ə o
ɛ ʌ ɔ
  a

/ie/ is the only diphthong. The central vowels /ɯ/, /ɚ/ and /ʌ/ do not occur in open syllables.