E.g., what is the right pronunciation of the word earn -- [ɹ̩n] (syllabic R) or [ʔəɹn] (glottal stop + schwa + R)?
EDIT: Is the word-initial (or more precisely "utterance-initial") syllabic R truly realized as a syllabic R or as a schwa + R (e.g., [ɹ̩n] or [əɹn])?
Starting a word-initial vowel (or syllabic consonant, as the case may be) with a glottal stop is a phenomenon known as hard attack. Geoff Lindsey has an interesting video about this, and his book English After RP also covers it (pp. 91-93).
Essentially: in traditional British Received Pronunciation (and also, I think, in older American dialects), hard attack was rare, used occasionally to emphasize the word preceded by the glottal stop. More recently, though, some speakers have begun to adopt hard attack much more widely. This has become particularly common in American English when the words the and to precede a word-initial vowel. There some speakers, instead of using the forms /ðɪj/ and /tʉw/, will use the normal pre-consonant forms /tə/ and /ðə/, using hard attack to separate the /ə/ from the following word. (For my part, I would use hard attack in to earn, saying something like [tʰəˈʔɹ̩̈n], but generally not in other contexts.)
That said, the more widespread use of hard attack is, as is stated in English After RP, "much more common in ‘presentational’ speech such as newsreading, where speakers are under special pressure to be clear, than in conversation." For this reason, he still suggests that EFL students try to avoid adding these extra glottal stops if their native language includes them.
The other part of your question concerns whether to use a syllabic [ɹ̩] or to start the word with a schwa [əɹ]. Both can be found, though this page from UManitoba claims that the syllabic consonant is much more common, and I would agree. (That said, there may be an exception when the word is itself preceded by an /ɹ/; I think that most speakers would, when saying a phrase like for earning, insert some sort of vowel between the end of for and the start of earning to keep those sounds separate.)
The initial glottal stop is not a phoneme in American English — you can pronounce earn with or without a glottal stop in front of it, and it will be equally correct, and probably equally well understood.
English is not like German, where you have to put a glottal stop in front of any word beginning with a vowel.
Generally we run our words together, but sometimes, especially if a word starting with a vowel follows another word ending with a vowel, or if we want to emphasize a word, we put glottal stops in between to separate the words. So the same speaker will very likely pronounce earn with or without a glottal stop in front of it, depending on its phonetic environment. Different speakers do this to different extents.
I don't know what your native language is, but you seem to have adopted a feature of German pronunciation in your English.
The quick answer is: both versions occur, depending on context, but the more basic form is the one without the glottal stop — this is standard in most contexts, and not glaringly weird even in contexts where adding the glottal stop is more standard.
Long answer: There are two separate differences here — the presence/absence of a glottal stop, and the use of syllabic r [ɹ̩] versus schwa+r [əɹ]. These are largely independent, so I’ll take them separately. Like OP I’ll take earn as the example, but this all holds for other words beginning similarly.
The glottal stop is most interesting: It depends on the context around the word earn. The basic pronunciation of the word for most AmE speakers is [ɹ̩n] or [əɹn] (no glottal stop). Most speakers will add a glottal stop (“hard attack”) before it in certain contexts — most commonly when earn follows a word ending with a schwa, or begins a new utterance, or is especially emphasised — and depending on dialect, different speakers may use hard attack in a narrower or wider range of such contexts. alphabet’s answer gives a more detailed and well-referenced discussion of this, so I’ll just illustrate with a few examples:
The version without glottal stop is more basic (“unmarked”, to be more precise), so if in doubt, that’s the safer one to use; omitting the glottal stop where a native speaker would use it is much less noticeable than adding it where a native wouldn’t. (This point is very tricky in ESL teaching, since when a native demonstrates the individual word earn, they’ll usually add the glottal stop, as it’s both emphasised and utterance-initial. It’s the natural pronunciation for the context, but it’s not the standard unmarked pronunciation of the word!)
I’m less sure about the second difference, [ɹ̩] vs [əɹ]. My impression is that both forms are widespread, and the difference is mainly inter-speaker or free variation (not context-dependent for individual speakers), and much less salient to native speakers than the presence/absence of the glottal stop.
(alphabet’s and Peter Shor’s answers each give very good further detail on specific aspects of the issue.)
As there isn't a phonemic contrast, it would not be surprising if some speakers use [ɹ̩n] while others use [əɹn]. My impression is that a syllabic consonant is the most common pronunciation; this is supported by the following quotation from a 1982 article by the linguist Arnold Zwicky:
For most American English speakers, the sound at the beginning of ermine is articulated just like the sound at the beginning of radio; for these speakers, ermine does not PHYSICALLY begin with a vowel followed by r. Yet the r at the beginning of ermine, like the r in the middle of bird and the r at the end of butter, counts as making a syllable, while string has a radio-type r, and only one syllable.
("Phonemes and Features", page 62, originally published in Innovations in Linguistics Education 2.2)
As other answers have covered, the use of a glottal stop at the start of words like this is optional.
