This text lists three sources (Daniel Jones, J. D. O’Connor and A. C. Gimson) about the English diphthongs in British English/Received Pronunciation. This is a simplified classification and by no means complete.
For greater details about the English diphthongs see this post.
The most detailed overview gives Jones (1975), who lists the important and less important diphthongs. The “essential diphthongs” (98) are: ei, ou, ai, au, ɔi, iə, ɛə, ɔə, uə; including “rising” ones: ĭə, ŭə, ŭi. Jones than continues with a note that two of those diphthong can be ignored by foreign speakers: ɔə and ŭi. The diphthong ɔə because it is replaced by ɔ:, and ŭi since “it can always be replaced by disyllabic u-i”. “(…) Nine further non-essential diphthongs”, according to Jones, are: oi, ui, eə, aə, aə, oə, ŏi, ĕə, ŏə. They can be replaced by their “fuller forms”.
Thus Jones gives 10 important diphthongs: ei, ou, ai, au, ɔi, iə, ɛə, uə, ĭə and ŭə. (They are not in slash parentheses; they are left as in the original text).
The next author is J. D. O’Connor (1973), who list 9 diphthongs (153): /eɪ/, /əʊ/, /ɑɪ/, /aʊ/, /ɔɪ/, /ɪə/, /ɛə/, /ɔə/ and /ʊə/. He then continues with the explanation that /ɔ:/ and /ɔə/ are not separated in pronunciation (“relatively few RP speakers make a contrast”) , so /ɔə/ is not essential.
In O’Connor’s division there are 8 essential diphthongs in British English (RP), as shown above.
Gimson (1970) refers to diphthongs as “diphthongal vowel glides” (126), and lists 8 of them (pp. 127 – 144): /eɪ/, /aɪ/, /ɔɪ/, /əʊ/, /ɑʊ/, /ɪə/, /ɛə/ and /ʊə/. Gimson goes at great depths and analyses each of the diphthongs, further explaining their long and short forms, as well as the variants.
Thus, according to Gimson, there are 8 significant diphthongs in RP English.
If you need to search for diphthongs in a specific context, take a look at FONRYE program that searches a phonetic dictionary.
To see the sources for the above, visit Books & References page. You can also read a discussion about drawing a diphthong by using its formant frequencies (and calculating the Euclidean distance).