I've seen the "-er" sound in English (like in butter) transcribed in all three of the above ways, but I've heard there are subtle differences between them.
What are these differences, if there are any?
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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.
[əɹ] denotes an [ə] vowel (schwa, or the mid-central vowel) followed by an [ɹ] (alveolar approximant, or "English R") sound. Key is that you can hear the transition from the vowel to the consonant, such as in this audio example.
But in practice these two sounds often melt into one in US English. The [ə] is rhotacized, also known as r-colored, resulting in [ɚ]. (See the Wikipedia article titled "R-colored vowel". It has an audio example comparing [ə] with [ɚ]) Basically this means you pronounce an English R with your mouth in the [ə] position. It is one sound. Here is an audio example where you can hear it.
The diacritic under ɹ in [ɹ̩] means the sound is syllabic. In other words, it is an R pronounced as its own syllable, as opposed to being part of a syllable preceding or following it. I don't know exactly how this is different from [ɚ] if at all, but my first impression is it might suggest the mouth does not have to be in the [ə] position. I did some Googling, and according to a few random folks on the internet it is actually synonymous to [ɚ] or [ɝ], but I have not seen a credible source yet so do not take their word for it just yet.
So in short: [əɹ] is ⟨er⟩ pronounced as two separate sounds while [ɚ] and [ɹ̩] are when it's pronounced as one sound.
In English phonology (esp. American English), these are all different ways of transcribing/representing the same thing -- a rhoticized schwa sound -- when using a broad (phonemic) transcription.
/ɹ̩/ transcription is similar to other syllabic consonants, where you may see them transcribed with either a schwa before them, or with a syllabic diacritic. Some transcriptions (e.g. the OED) even leave out the syllabic diacritic, although for LETTER (Wells' lexical set), they use
ə(r) to denote it being rhotic in some accents.
I have found
[ɹ̩] used in the
us1 MBROLA text-to-speech voice (as
ɜː [NURSE] and
ɚ [LETTER] (as American English does not to distinguish the two).