What are the 15 vowel sounds in English?

There's a slight misconception in the question. English actually has 20 vowel sounds (not 15), which are represented by a combination of letters and sometimes digraphs (combinations of two letters).

Here they are, along with example words:

Short Vowels:

1. /ɪ/ (kit)

2. /ɛ/ (bet)

3. /æ/ (bat)

4. /ʌ/ (but)

5. /ɒ/ (cot)

Long Vowels:

6. /iː/ (beat)

7. /eɪ/ (bait)

8. /ɑː/ (bart)

9. /ɔː/ (bought)

10. /ʊ/ (book)

Diphthongs (Vowel glides):

11. /eɪ/ (say)

12. /aɪ/ (buy)

13. /ɔɪ/ (boy)

14. /əʊ/ (go)

15. /aʊ/ (cow)

Other Vowel Sounds:

16. /ɜː/ (bird)

17. /ə/ (about)

18. /ɪə/ (beer)

19. /eə/ (bear)

20. /ʊə/ (tour)

It's important to remember that these are just the sounds, not the spellings. Many different letter combinations can represent the same sound.