Apart from the problem of defining the specific dialect you are interested in, you also need to be more specific about what theory of "phoneme" you are assuming. The main theoretical split is between those who just look at narrowly transcribed surface pronunciations and reduce variation to phonetically-defined rules, versus those who construe "phoneme" as being "sound that cannot be derived by rule". The contested area is when a vowel quality is not phonetically predictable, because a conditioning factor changes (e.g. r is later deleted, schwa is later inserted). There will always be some unresolvable controversy about length / tenseness / diphthongs (i.e. does English have the contrast /[i:], [ɪ]/ or [ɪ]/[ɪj] – there are other analyses, so expect some chaos).