While what @Raizin says about [əɹ] is technically true--it is supposed to denote a sequence of two phones--I have seen [əɹ], [ɚ], [ɹ̩], and [ɝ] all used to refer to the same speech sound. The thing is, while schwa is given its own place in the vowel space chart, in practice the formants (other than the first formant) of schwa tend to just be transitions between the consonants surrounding it (unless it is phrase-initial or phrase-final), so it's more of a chameleon vowel, blending into its surroundings. Consequently, a schwa-r sequence is going to be virtually indistinguishable from an 'r-colored schwa'. Furthermore, one would be hard-pressed to distinguish between unstressed [ɝ] and [ɚ].

I think, more often than not, it's a matter of convention rather than of phonetic distinctions. For example, I know one phonologist who sticks to [ɝ] and [ɚ] for transcriptions of American English, and by convention she uses [ɝ] in stressed positions (as in 'sir') and [ɚ] in unstressed positions (as in 'ulcer'). It's true that the formants of the vowel in 'ulcer' are going to tend to be a bit less extreme (i.e., a bit more centralized) than those of the vowel in 'sir', especially at faster speech rates. But they aren't always, and yet she lets phonology, not phonetics, dictate which symbol she uses. This follows the convention of transcribing the vowel in the unstressed syllable of 'submit' as [ə] and that in the stressed vowel of 'substance' as [ʌ], even if instances of those respective words can be found in which the formant values of the vowel are identical in the two cases.

While what @Raizin says about [əɹ] is technically true--it is supposed to denote a sequence of two phones--I have seen [əɹ], [ɚ], [ɹ̩], and [ɝ] all used to refer to the same speech sound. The thing is, while schwa is given its own place in the vowel space chart, in practice its formants (other than the first formant) tend just to be transitions between the consonants surrounding it (unless it is phrase-initial or phrase-final), so it's more of a chameleon vowel, blending into its surroundings. Consequently, a schwa-r sequence is going to be virtually indistinguishable from an 'r-colored schwa'. Furthermore, one would be hard-pressed to distinguish between unstressed [ɝ] and [ɚ].

I think, more often than not, it's a matter of convention rather than of phonetic distinctions. For example, I know one phonologist who sticks to [ɝ] and [ɚ] for transcriptions of American English, and by convention she uses [ɝ] in stressed positions (as in 'sir') and [ɚ] in unstressed positions (as in 'ulcer'). It's true that the formants of the vowel in 'ulcer' are going to tend to be a bit less extreme (i.e., a bit more centralized) than those of the vowel in 'sir', especially at faster speech rates. But they aren't always, and yet she lets phonology, not phonetics, dictate which symbol she uses. This follows the convention of transcribing the vowel in the unstressed syllable of 'submit' as [ə] and that in the stressed vowel of 'substance' as [ʌ], even if instances of those respective words can be found in which the formant values of the vowel are identical in the two cases.