Steve Bett’s April, 1999 Simpl Speling Newsletter article 
Can we pin down the number of phonemes in English?
What is the minimum phoneme inventory for English?
This article became the inspiration for a much more detailed description of the problem of linguistic representation published in the Journal of the SSS.  Bett thought that there could be agreement on the number of pure phonemes. The later article cast doubt on this pointing out that different dialects of English may require different numbers of phonemes

In her letter to the Daily Express, Masha said: "We have 256 ways of representing the 45 basic sounds of our language." According to Jean,  Diane McGuinness (1998) gives the figure for American English as 42. In Alan Campbell's letter to The Press, he said, "There are 41 sounds in English and almost 600 ways of spelling them." In PV7, Steve said, "There are 41 significant speech sounds or phonemes. "In a phonemic or alphabetic system, these sounds would be spelled about 41 ways. "In the traditional English writing system they are spelled over 500 ways." Godfrey Dewey (1971) listed examples of 561 ways that 41 English sounds could be spelled. 

Allan asked, "Is there any way to pin down these statistics so there is some consistency in our public claims." The quick answer is probably not. While we can be specific about the minimum number of pure (uncombined) phonemes required to fully describe English speech [34], it is nearly impossible to reach agreement on the number of  phonemes when combinations are included. The key reasons for this are ... 

  (1) There is no obligation for a particular orthography to list any combination or blend.
  (2) An orthography that listed every combination or blend used in a transcription as a
       separate phoneme would have over 60 phonemes.

A phoneme is a range of sounds that are treated as equivalent by a speech community. It is the smallest unit of sound in the sound system of a language that can make a difference in meaning.  The phoneme inventory for English was charted over 100 years ago by Pitman, Ellis, Jones,  Sweet and others. They were all searching for the minimum number of phonemes required to graphically represent southern educated English speech sometimes referred to as BBC English or RP. They all agreed on 34 pure phonemes.  The number with combinations, however, varied from 40 to 50. 

The Longman Dictionary of American English  recognizes 45 phonemes (21v-24c).  Longman recognizes schwa but merges (a: and o) since American's do not make a distinction.  Longman does not single out the combinations [hw] and [yu:]. Five of Longman's phonemes are r-combinations. If these are eliminated, the number of significant phonemes is reduced to 40.  40 is the most common number of isolated sounds in popular notations and pronunciation guides   [see Spelriet, Unifon,  and  Truespel]. 
 

25 Vowel Phonemes for English - 12 pure 3 Notations:  IPA - Chkt Spl - Unigraf downsize 6 Chekt - short 6 Free - long Difthongs 4-7 with schwa æ . .a.  .a a:. a . .q ai.'y .I a'.arqr |ai' 'yr Ir at, ax, ask,  cat alms, want, watch eye, ice, bite are, car    |  ire, fire e   .e .e 3:.'r . . R ei . .ei  . A e' . .er  . er edge, edit, elbow her, girl, skirt, urban ace, ape, vein air, care, there, barely i .. .i.  ..  i i: .  i . . E oi . .oi   . Q i:'  .  ir   . ir it, in, index, ill eel, east, mere, very oil, boy, loyal ear, fear, deer, pier o turned ao. q o: turned c .o.o 'u .o' . O or . .or  . or ox, otter, cot, pot awe, call, cost, loss,  oh, oat, low for, four, floor, more u . .u. . .c .J u: .u  . .C  V ju .iu .U ju'. .iur  . Ur hook, put, would ooze, zulu, zoo, duty you, few, fuse your,  sure |  tour, poor L   .u'  . u 'turned e .a' . u au.au M auc .aur  Mr up, cut,a'bu't a'kUt ago, sofa, atom, silent out, down, cow our, flower, power IPA's turned e, a, v, and c are unavailable in ASCII and Latin-1 Darker shaded cells not members of Jones' minimum phoneme set Chart as Grahic  IPA chart

The 12 pure vowels are shown in the first two columns.  The first notation is IPA -International Phonetic Alphabet. The 2nd is Chekt Speling. CKS marks the checked vowels with a dot contractor instead of a colon extender. Since checked vowels are always follwed by a consonant the dot does not get confused with punctuation. White cells indicate phonemes that are sometimes merged.

The first two columns in the chart list the 12 pure vowels (6 checked - 6 free). Chekt vowels are always short and always followed by a consonant.  Descriptive orthographies include all 12 pure vowels. Pragmatic notational systems, such as Unigraf, may merge similar sounding phonemes such as the central vowels.  In Unifraf and Truespel notations, the lower case  u refers to both /L/ and /'/ and the open back vowel [ah] is associated with one symbol rather than two [a and o].    [o. and a:] = q in Unigraf and aa in Truespel.

Almost everyone agrees that the consonant combinations tsh [ch] and dzh [j] and the diphthongs ei ai ou au and oi are essential.  When these 7 sounds are added to the 12 pure vowels and 22 pure phonemes, the total number of essential phonemes climbs to 41.

Sweet and Jones add 4 schwa combinations yielding 21 essential vowels and a phoneme inventory of 45. In their transcriptions Jones and Sweet used more than the 45 phonemes found in their inventory. Their list did not include combinations with consonants [ju] or triphthongs [ai@ and au@].  Chekt Spelling adds [iu] and 3 more schwa combinations to bring the total number of vowels to 25 and the total number of phonemes to 50. 

The minimum number of pure phonemes required to accurately transcribe English speech is 34 (12 vowels + 22 consonants). Pragmatic orthographers have frequently chosen to merge [a:/ o],  [u/shwa], and [th / dh] reducing the number of phonemes by 3.  TrueSpel and Globish also ignore [ng].  With these phoneme mergers, the number of symbols required drops to 30.  

Eliminating any pure vowel tends to distort the description of RP and most other variants of English. However, a carefully pruned 30 phoneme version of English would still be intelligible. 

A complete broad representation of spoken English requires 34 pure phonemes.  When commonly used combinations such as J, Ch, ei, ai, ou, au, and oi are added, the number of essential phonemes (not counting R-combinations) in a descriptive orthography is 41. 

Article in JSSS
Sweet

It is true that different dialects of English have different phoneme counts.  This does not preclude the development of a broad notation for all dialects.  A sound that is phonemic in one dialect but not in another would have to be included.  American English rhymes bother and father making it possible to merge a: and o in pronunciation dictionaries for American English. 

 http://hawk.hama-med.ac.jp/dbk/new_romaji.html