I count eighteen rhymes in Dirty Old Town - five consonance rhymes (repeating consonants), eight assonance rhymes (echoing vowel sounds), and five alliteration rhymes (repeating initial letters). Whilst the rhymes might not all be intentional, a lyric writer with a good feel for language and diction will naturally turn out subtle rhymes. By way of comparison, The Streets of London (also four verses long), would actually have less rhymes, unless you count the repeating chorus four times.
Just saying…..