As far as I know, no one has ever really addressed this question. I replied
In using the International Phonetic Alphabet, the only constraints on combinations of diacritics and base characters are logical ones. For example, the “voiceless” diacritic U+0325 COMBINING RING BELOW can logically only be applied to a base character that stands for a voiced sound, e.g. [m, b, ɓ, u], but not [t, θ, ç, ʘ]. It has a variant, U+030A COMBINING RING ABOVE, used if the base character has a descender, e.g. [ŋ, ɻ, ɡ]. Similarly, the dental diacritic U+032A COMBINING BRIDGE BELOW would usually only be applied to a base character standing for an alveolar sound, e.g. [t, n, s]. However if someone wanted to use it with, say, [b] to show a labiodental stop, that would also be OK. But combined with, say, [k] it would presumably be meaningless.
There is no formal constraint on having multiple diacritics on the same base character.
The aspiration diacritic, [ʰ], is most commonly deployed after symbols for voiceless plosives. The IPA Handbook also shows it with [d], although the article about Hindi in the body of the book instead uses [ʱ] for the voiced aspirated series, thus [bʱ d̪ʱ dʒʱ ɖʱ ɡʱ]. Korean has two kinds of alveolar fricative, one of them often transcribed [sʰ], and I suppose in principle any affricate or fricative can be aspirated. Can approximants? Can vowels? You could call the preaspiration of Icelandic, Scottish Gaelic etc aspiration of the preceding vowel, and certainly transcriptions of the type [kʰaʰt] are in use for such cases.
“Breathy voiced” and “creaky voiced” can only combine with symbols for voiced sounds: [t̤] is presumably a logical contradiction. Nasals can’t be nasalized. And so on.
Are “advanced” and “retracted” only for vowels? No, because they are sometimes used to show dental as against alveolar, prevelar as against velar consonants. What about “centralized”? Vowels only, I think. Is “syllabic” only for consonants? Normally yes, and then only for nasals and liquids. Some students imagine that looked should be transcribed lʊkd̩ (with “syllabic d”), but they are confusing phonetics with morphology. Syllabic plosives are a no-no.
What about combining the syllabicity mark with a vowel symbol? Normally we don’t do that, because vowels are inherently syllabic, so it would be tautologous. But Abercrombie, in his English Phonetic Texts (1964), used a transcription for English in which the syllabicity mark sometimes appeared under schwa to show that it was not part of a diphthong.